THE UN I VERS ITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 21 SOBOLEV FUNCTIONS AND SOBOLEV SPACES Lamberto Cesari ORA Project 024160 submitted for: UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1971

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ADDENDUM VII. SOBOLEV FUNCTIONS AND SOBOLEV SPACES VII 1. THE TEST FUNCTIONS Given any open subset G of the t-space E, t = (t,...,t ), v > 1, we shall denote by Co (G) the family of all real valued functions p(t), t C G, which possess (continuous) partial derivatives of all orders in G. We shall denote by C (G) the family of all c c C (G) with compact support K c G. The support K of a function cp(t), t E G, is the closure of the set of points t e G where cp(t) / 0. The following examples are of some interest. For v = 1 let us define a(t), - < t < +o, by taking a(t) = 0 for It, > 1, and a(t) = exp (t -1)-1 for Itj < 1. Then a E C (El). If we take C(t) = c t a(T) dT, -a < t< t, then a(t) = 0 for t < -1, and by choosing the constant c > 0 suitably we can arrange that a(t) = 1 for t > 1. Then, 0 < a(t) < 1 for all -1 < t < 1, and a c C (El). Finally, given any two numbers rl, r2, 0 < r1 < r2 < ~+ let a let us define B(t), - o < t < +oo, by taking -l +1 t (t;r,r2) = 1- u(-l + 2(jtj -r )/(r2-rl)). mem 1 1 t Then, P(t) = 1 for Itt < rl, P(t) = 0 for Itl > r and 0 <:(t) < 1 for r < Itl < r, and < e _ r2 _rl r l r2I co (EZ).

Now, given any open set G c E, v > 1, and any point t E G, let us take any number ~, 0 < ~ < dist (t, 3G), and let us define cp(t), t c G, by taking cp(t) = a(El It - t ). Then cp(t) = 0 for It - tl > e, cp(t) > 0 for It - t; < ~, and cp E C (G). Analogously, let us take any two numbers rl, r2, 0 < rl < r2 < dist (t, j2), and let us define *(t), t c G, by taking 4(t) = P( It -,l; r, r2). Then X(t) = 1 for t G G with It - tl _ r1, 2t) = 0 for t 2 G with t - > r, 0 < (t) < 1 for rl < It - tl < r2, and f E CC' (G). We shall denote by multiindex a = (a,...,Cv) any set of v nonnegative integers al".', and by Ij!J the integer jal = a1+.'.+. If CP E C (G), we shall denote by Djcp the partial derivative cp/dt, j = l,...,v, and in general by D cp the partial derivative of order a = (a1. a ) of c in G, a -V1 Va D p D DVvCP

VII 2. GENERALIZED DERIVATIVES Let G by any open subset of the t-space E, t = (t,...,t ),v > 1. As usual we say that a real-valued function x(t), t e G, is locally integrable loc loc in G, and we write x E L (G), or x E L1 (G),if x E L(K) = L (K) for any compact subset K c G. Analogous definitions hold for L -integrability, 1 < p < +,c, and even for p = +oo, where then we understand that x is essentially bounded in every K C G. If x c L (G), 1 < p < p < +, then by fIxilpG we mean the number p lxixipG = [f Jx(t) IPdt]l/P. If x E L (G), that is, x is essentially bounded in G, then ||lx|iG = Ess Sup Ix(t)| for t e G. b ic Given any function x(t), t E G, x E L (G), we say that another function y(t), t E G, y E L (G),is the first order generalized partial derivative of x with respect to ti provided fG y(t) p(t)dt = -JGx(t) (ap/ati)dt for all c E Ca(G). (VII 2.1) Analogously, given any multiindex a = (al,''', c), we say that a function y(t), loc t c G, y E L (G), is the generalized derivative of x of order a = (al,..'., ) provided JG y(t) c(t)dt = (-1) IJG x(t) D C(t)dt for all p c Co(G). (VII 2.2) 3

As we shall see these definitions generalize the usual concepts of differentiation, and hence we shall use the same notations y = D.x, or y = D x. 1 With this notation (VII 2.2) will take the symmetric form fG cp(t) D x(t)dt = (-1) 1 IGx(t) D (t)dt for all p E CC(G) (VII 2.3) (VII 2.i) If v = 1, if x(t), a < t < b, is AC in every closed interval loc [a, b] c (a, b), then x, x' E L (a, b), and x'(t) is the generalized derivative of x in (a, b). Proof: If cp E C (a, b), then the compact support K of cp is contained in some closed interval [a, b] c (a, b), and by integration by parts with Lebesgue integrals we have b J x'(t) c(t)dt f x'(t) y(t)dt = x(t) p'(t)dt a a a -= - x(t) p'(t)dt. a For t = (t,...,t ) E E and i = l,...,v, we shall denote by t' the (v-1)V 1 1 i il 1 (t 1,..t, t,.., t). If I = [ce, i] is an interval of E, 1 t 1 v 1 v i i 1Y v)O... ~. (,f ~v <, i = l,..~,v, and i = 1,...,v, we shall denote by I' the (v-l)-dimensional interval Ii [aS < t < P, s s i, 1 i1 v 1 i-i i+l v s = l,...,v]. If dt = dt...dt, we shall write dt' = dt.dt dt i.ldt We shall denote by G. the projection of G on the t'-space E that is, i i'i the set of all t' E such that (t', t i) G for some t i E1. Then G is 1 v-1 i 1 an open subset of E1, i = 1,...,v. Also, for every t' C G. we shall denote 4

by Gi(ti) the open subset of all t E E1 with (t, t ) G. (VII 2.ii) For v > 1, if x(t), y(t) E L (G), and if for a given i = l,..,v, and almost all t' E Gi, the function x(ti, ti) of t alone is AC in every i I1i closed interval [a, b] e G.(t'), with derivative y = ~x/at, then y is the generalized partial derivative of x with respect to ti in G, or y = ax/ti = Dix, i = l,...,v. Proof: To simplify notations we shall limit ourselves to the case v = 2, and we shall write (t, s) instead of (t,...,t ). Let us prove that ff y(t, s) c(t, s)dtds = -ffG x(t, s) (cp/pt)dtds for all cp e C(G). (VII 2.4) Note that, if K is the compact support of cp, then both x and y are in L(K L(K ) and cp and cg/at are continuous on K, hence yrp and x(adp/t) are integrable on K, identically zero on G - K, and finally L-integrable in G. Let G be 1CP T~0 the projection of G on the s-axis, that is, let G be the set of all s E1 such that (t, s) E G for some t, and thus G is open. For every s E G let 0O G(s) denote the set of all t with (t, s) c G for some t; hence G(s) is open, and therefore the union of countably many disjoint intervals (a, P). If K (s) has analogous definition, then the intervals (a, P) form an open cover of the compact set K (s); hence, finitely many of such intervals (a, ) cover K (s). (P (p By Fubini's theorem we have JJ Y J dtds = JG dsG( Y dt= ds(

and an analogous relation holds for x(3cp/<t). Here Z ranges over the finitely many intervals(ac, ) above with (Ca, p) n K(s) / 4. If [Co, a ] is any closed interval with (a, I) D [c, o] o (a, ( ) n K (y), then cp(a, s) = (P, s) = 0, and by integration by parts | y cp dt = y C dt= -| x(//t)dt -It)dt 0 0 This holds for any s E G such that x(t, s) is AC with respect to t on each [ao, i ], that is, for almost all s e G. This proves that fJG y p dtds = fG ds( c y f dt) = -SG ds(x ft G G fyGPdt) -f ds(Zf x(ap/at)dt) 0 0 IG x(b cp/t) dtds, that is, y = 8x/ti is the generalized first order partial derivative of x i with respect to t according to the definition (VII 2.1). In particular, if x,y E L (G), and x is continuous in G together with its usual partial derivative y = ~x/at (or x,y e L (G)n C(G)), then y = x/2ti = Dix is the generalized first order partial derivative of x with respect to t in G according to (VII 2.1). Analogously, if x(t), t E G, is Lipschitzian in G(or on every compact subset K of G, then the usual first order partial derivatives D x(t), i = l,...,v, of x, which exist a.e. in G and are measurable in G, and bounded in G (or on every compact subset K of G), are also the generalized first order partial derivatives of x in G. A statement analogous to (VII 2.ii) for partial derivatives of order m is

as follows: (VII 2.iii) For v > 1, m > 1, if x is continuous in G with usual continuous partial derivatives of all orders < m - 1, if for each a with Ia! = m - 1, and almost all t: E G., the function D x(t' ti) of ti alone is AC in every closed 1 1 1 1 interval [a, b] C G (t'), i = l,...,v, then the usual partial derivatives of 1 1 order m certainly exist a.e. in G. If these derivatives are known to be in L (G), then all usual partial derivatives D x of orders 0 < Iaj < m are also loc generalized partial derivatives (and all are in L (G)). A corollary of (VII l.i) and (VII 2.ii). Formula (VII 2.4) need only be applied lal times. A first remark concerning generalized partial derivatives is that if y is the generalized partial derivative of x of some order a, then the same holds for any other two functions y and x, where y differs from y and x from x at most in sets of measure zero in G. In other words, the relation between x and y defined by (VII 2.2) is a relation between the equivalent classes in Lebesgue integration theory defined by y and x. (VII 2.iv) If two functions Yl1 Y2 C L (G) are such that fG Y1l pdt = G Y2 cp dt for all test functions cp, then Y1 = Y2 a.e. in G. In particular, generalized derivatives in G, if they exist, are uniquely defined (up to a set of measure zero in G). 7

Proof. We have here G (Y1 - Y2) cp dt = 0 for all q0 E Co(G). Since - Y2 E Lc (G), then for almost all points t E G we have Iq-1 JG (Y1 - Y2)dt Yl() Y2( as ~ + O, where q is the sphere of center t and radius ~ > 0, and t E q c G. For every such t and > 0 such that t e q C G, we can well determine a number, 0 < a < 1, a = a(), so small that fq'-q lY1 - y21dt < ~Jql, t E q c q' c G, where q' is the sphere of center t and radius e(l + a) = s'. Now let us consider a function *(t; e, ~'), t E G, with 4 c C, 4 = 1 for It - tl < ~, 4 = 0 for t - et ~ ~', 0 < < 1 for ~ < It - tl < ~' (cf. VII 1). Then we have 0 = ( - Y2) dt = q, (Y1 - Y2) dt =q (Y1 Y2) Iql for some -1 < G < 1, and finally o = fG(Yl - Y2)(1- l )dt = Iql- q (Y1 - Y2)dt + G ~. As E~ 0 we have 0 = Yl(t) - Y2(t). This holds for almost all t E G. Thus, we have proved that Y1 = 2 a.e. in G. (VII 2.v) If x, y E L (G), if y = D x in G, and G c G, then y = D x in G. can be extended to an element 00 o

(or p E C (G)) by taking = O in E - G (cp = 0 in G - G). By ( we have now fG y p dt = G Y dt = (-1) afG x(D P)dt = (-1) I x(D P)dt, 0 0 (VII 2.4) and this proves that y D x in G 0 Example 1. We are now in a position to exhibit a function z(t), 0 < t < 1, (v = 1), which has no first order generalized derivative y = z' in (0, 1). Consider the usual function z(t), 0 < t < l, defined in association with the ternary Cantor set S in [0, 1]. Then, z is continuous and monotone nondecreasing in [0, 1] with z(O) = O, z(l) = 1, and z(t) constant on each interval (ca, A) of the open set G = [0, 1] - S, with JGI = 1, IS I = O. Let us prove that z possesses no first order generalized derivative y = z' in (0, 1). Indeed, assume that y is such a function. Then y is a generalized derivative of z also in each interval (a, A) where z is constant, and hence has derivative zero in (cx, 3). Thus, by force of (VII 2.i) we have y = 0 a.e. in each interval (c, a), and hence y = 0 a.e. in G, and y = 0 a.e. in (0, 1). By (VII 2.1) we have now ft z(t) qp'(t)dt = O for every Cp E C (0, 1). Now take any two intervals of constancy for z(t), say (ct, a) and (c','), 0 < c < P < c' < i' < 1. Then z takes on values c, c', respectively, 0 < c < c' < 1, in (a, P) and in (a', B'). Take any two intervals of the same length B, say [a, a+t] c (ac, B), [b, b+nr] c (cx', 3'). Take a function a(t), -oo < t < +oo, with a = 0 for t < 0, a = 1 for t > A, 0 < a < 1 for 0 < t < T, and a E C(-oo, -00) (cfr. VII 1). Finally, let us define ~(t), O < t < 1, by taking d(t) = O for 9

O < t < a, i(t) = 0 for b+1 < t < 1, *(t) = 1 for a+_ < t < b, i(t) = a(t - a) for a < t < a+r, J(t) = a(b +i- t) for b < t < b+n. Then i e C (O, 1), ~'(a + t) =-r'(b + ~ - t) =.(t) for 0 < t < A, i' = 0 otherwise, and finally (fa+,l + b+n tz 0 = o z(t) *'(t)dt = (I+ + b )' dt = (c - c')f|Y(t)dt < 0. a a b O a contradiction. Example 2. We can now exhibit a function x(t, s), O < t < 1, 0 < s < 1, (v = 2), posessing generalized mixed partial derivative a x/Otas but no first order generalized partial derivatives cx/at and cx/as. Let us take the same function z(t), 0 < t < 1, considered in example 1, and define x(t, s) by taking x(t, s) = z(t) + z(s), (t, s) E G = [O < t < 1, 0 < s < 1]. Let us prove that the generalized mixed partial derivative y(t,s) = 0 in G. Indeed, for y = O, we have II y(t, s) p(t, s)dtds = 0 for all cp E C (G). On the other hand, for every (p E Co(G) we have also 2 1 IIf y(t,s)( p/ata2s) = f z(t)dt fo 2 //tas)ds + f z(s)ds f|l( cp/tas)dt = 0 because the two interior integrals are both zero for every t and s, respectively. Thus (VII 2.2) holds for y = O and all cp e C (G); hence y = O is the generalized mixed derivative a s n the other hand, the same argument used in example mixed derivative 6 x/ctQs. On the other hand, the same argument used in example 1 shows that x has no first order generalized partial derivatives ~x/~t, ~x/6s. (VII 2.vi)(Partition of unity).. Given any compact set K and any finite open mc00 cover Ui=1 Gi of K, there are functions $i(t), t E Ev, i E Co(Ev), with 10

supp i C Gi, i = l,...,m, such that =l1 5 i(t) = 1 for every t E K. Proof. First, all we have to do is to construct suitable compact sets Ci, C, i = 1...m, such that K C Ui 1 C and C. C int C' c CI C Gi, i = l,...,m. To do this we note that every point t e K belongs to some Gi, and we can take a well determined sphere S(t, 28) of center t and some radius 28 such that S(t, 28) c Gi. Now the spheres (S(t, 8)) form a cover of K, and hence there is some finite cover S(tk, 8k), k = 1,...,N, and for every k = 1,...,N, we can shoose a well determined i, say i = i(k), such that S(tk;8k) c S(tk,26k) C Gi(k), k = 1,...,N. Now let Ci be the union of these spheres S(tk,8k) such that i(k) = i, and let C! be the union of the corresponding spheres S(tk,2 k) Then all sets Ci, C', i = l,...,m, as finite unions of spheres, are compact, U Ci M K, and C. int C' c C! c Gi, i = l,...,m For every k = 1,...,N, let cPk(t) = B(t-tk, 8k' 82k) be the function defined in (1.1); hence 1k * CO (E), ~k = 1 on S(tk,5k), k = 0 in E - S(tk,26k), O < 1 otherwise. Now we take 1 = c1, 2 1= (1-P1)P2P, 3 = (1-P1)(1-P2)P3, "P P 9N = (l-P1)). (1(PN-l1)(PN. It is immediately seen that N=l k = + (1-cP1)cP2 +.+ (l-0 Pl)..(lpNl1)N = 1 - (z-~)(-2).. (- 2N) for all t E. On the other hand, k C CO(EV), k > 0 in E, and k=1 k = for every t c K, since t c K implies t E S(tk,8k) and ck = 1 for at least one k. Finally, for every i = l,...,m, let *i(t) = Z k(t), where Z ranges over all k with i(l) = 1. Then x=l ti(t) = 1 for all t 6 K, ii 6 C (E), f.i > 0 on Ev. 11

Also, the support of ri is contained in the union of the spheres S(tk,2Sk) with i(k) = i, all these spheres are in Gi, and supp ri c Gi, i = l,...,m. Statement (VII 2.vi) is thereby proved. We are now in a position to prove a statement which is essentially the converse of (VII 2.ii). loc (VII 2.vii) If x, y e L (G), if there is an open covering (r) of G such that y is the generalized derivative of order a of x in each r, then y is the generalized derivative of order a of x in the whole of G. Proof. Let cp be any element of C (G) with compact support K. Let G o-q) o be an open set with compact closure such that K c G C cl G c G. Then (r) is an open cover of cl Go, and hence there is a finite subcover, say [rs, s = 1,...,N]. By force of (VII 2.vi) there are functions A s(t), t e G, with compact support Ks C r such that' E Co(G), s = 1,...,N, and such that Z (t) = 1 on cl G. Now we have s=1 s o f y cp dt= f y cp dt= s=l N G cl G cl G Y( -S=1 )cp o o N N s =1cl G Y ts cP dt= s=lIF y(r ~s)dt o s where c s E CO(r). Hence s o s) f y(cp t )dt = (-1)la I x(D ( ))dt because of the property of y to be the weak derivative of order ca of x in each r. Finally, fG Y c dt = (-1)aIZ a I fGy s dt =(-1) I( f x( D C SS D cp)dt o 12

Since ZsJs - 1 in G, all derivatives Dv (Zss) with B > 0 are zero in G and G y cp dt = (-1) G x (D P)dt = (-1) IG x (D p) dt. 0 Statement (VII 2.vii) is thereby proved. (VII 2.viii) (Leibnetz rule). If x E L (G), 1 < p < +00, y E L (G), p q a 1 < q < +Co, 1/p + l/q < 1, possess generalized partial derivatives D x, D y oLloc,loc of all orders < al < m, and of classes L (G), L (G), respectively, and P q 1 < X < +o is so chosen that 1/p + l/q _< l/, then xy e L (G) possesses generalized partial derivatives D a(xy) of all orders 0 < jal < m, all in L (G), and they are given by the Leibnitz rule aDa( (XY) = ) D x Da y a.e. in G, (VII 2.5) where a = (e,...,a ), 0 < a m, P = (Pl,., "v)'V where Z ranges over all 0 < I < a, that is,0 <. < a< i i = l,...,v, and P) a:= /'(a- C ) a..a "/ Cy' " a ( l- P )' "(av - P) (ffi) "E /f3 ((y ) 1 v 1 v 1 1) v v Proof. It is not restrictive to assume (A/p) +(X/q) = 1. Then x E L p/(K), y E L q/(K) for every compact subset K of G. By H'lder inequality with exponents p/X, q/X we have ( IxylX dt < (fKlXlP dt)k/P (fKjy)q dt) /q loc Thus xy E L (K) for every compact K c G, and xy E L (G). The same holds for 13

each of the products D X D y which appear in (VII 2.5). First, let us prove (VII 2.5) for y E C (G). Let cp be any element of 0 C (G) and note that G ( e ()DX DU-P y)m dt = i (f) IG(D x)((D -y)p) dt = Z(-1) B(aB) iG x DB((D y)cp)dt ~ G JG X[z (-1) ( )D DpD y)dt, (VII 2.6) where Z ranges over all 0 < D < a and Z over all 0 <7 < <. In these transformations we have applied the Leibnitz rule on the products (D y)cp of functions of C(G). For y = a we have y = = a and the corresponding term in brackets is (-1) ly D p. The remaining terms in brackets have sum zero because of the identity ZC(-l1) l($)(y) = O. Thus f a -BF Ice fG(E (P)D x D Y) cp dt (-1) IG(XY) D C dt (VII 2.7) for every p E Co(G). Thus, by force of (VII 2.iv), the derivative D (xy) exists and is given by (VII 2.5) a.e. in G. We have proved relation (VII 2.5) for y E C (G), 0 < Jaof < m. Now let us assume y E L (G). The same argument above applies since now q we can use the Leibnitz rule on the products (D -y)cp and partial derivatives of orders 0 < y < B which appear in (VII 2.6). We obtain thus relation (VII 2.7), which shows that DC(xy) exists, 0 < Ia <in m, and is given by formula (VII 2.5) again by force of (VII 2.iv). 14

Loc a c oc (VII 2.ix) If x E L (G), if y = D x Ll~(G), and z = D y E L (G), p P P then z = D a+x. Proof. For every p E C (G) we have fG x(D p+)dt = x D (D p)dt = (-1) If(D X)(Dcp)dt = (-1) fG y(D cp)dt = (-1) + (D y)cp dt Ia+B I = (-t1.) I fG z cP dt. This proves that z = D x. (VII 2.x) Given functions y (t), t E G, for every a = (,...,a) with o < Ia M < m<, y E L (G), and a sequence of functions xk(t), t E G, k = p k 1,2,..., possessing generalized derivatives D xk(t), t E G, DaXk E L (G), k = 1,2,..., such that, for every compact subset K of G, IIDxk - Yk II K 0 ask X- oo, then the function y = yo has generalized partial derivatives of all orders < m, and D y = ya in G, 0 < Ial < m. Indeed, if cp is any test function, with compact support K c G, then, for 0 < lal < m, we have a Iae a fG(D xk)p dt= (-1) IfG xk(D c)dt, k = 1,2,..., where the integrations can be made on K. By a passage to the limit on the integrals ranging on K, and then writing G again instead of K, we have IG Y cp dt = (-1)l IG y(D P)dt. Thus, y =D y for all O < a < m. 15

Remark. Statement above holds even if we know only that in every bounded open subset G with G c cl Go c G, we have D xk - y, as k ooweakly in Lp(G) (or even only weakly in L1(Go)). The proof is analogous. VII 3. MOLLIFIERS We shall consider any function j(t), t e E, of class C (E ) such that n 0 V j(t) > 0 for all t E E, j(t) = 0 for Itl > 1, and f j(t)dt = 1. An example of such a function is of course j(t) = c exp (iti2-1)-1 for Itl < 1, j(t) = 0 for Itl > 1, where c is a suitable constant (cfr. VII 1.1). For every ~ > 0 we shall now define j (t) by taking jE(t) = E- j(E -t). Then j (t) > 0 for all t E E, j (t) = 0 for itl > E, and fJE j(t)dt = 1. We shall now denote by J the operator defined by y(t) = (J x)(t) f j~(t-T)x(T)dT, G loc where x E L (G) and where y(t) is defined for all t E G with dist (t,aG) lOc > e. If x E L (E ), or x E L(G) for some G c E and we extend x to an element x E L (E ) by taking x = 0 in E - G. then y is defined for all t E E and y Lloc (E). lOc If x E L (G) and x has compact support K c G, then x E L(G), and even x x E L(E ) if x is extended to all E by taking x = 0 in E - G. Moreover loc (VII 3.i) If x E L (G) and has compact support K c G, then for every E, x o < e < dist (K,aG) we have y E C (G). x o The proof is left as an exercise for the reader. Below, we shall always assume that x is extended in E by taking x = 0 in E - G. With this convention, if x e L1(G), then y = J x is defined for v 1 all t E G (and even for all t E E ), and J: L1(G) C(G). 16

(VII 3-ii) If x E L (G), 1 < p < + oo, then y = J x E L (G) and IJi xjpG < p _ x pG - pG Proof. Let us assume 1 < p < + oo. By H5lder inequality we have I(JE )(t)IP = jf j(t-T)x(T)dTjP G 1/p l/q < IJ (j~(t-T))1/p x(T) ~ (j (t-T)) /qdT|P G where l/q + l/p = 1. Then, by Fubini's theorem we have IJEXIjp,G = ( I(J~x)(t) Pdt)1/P < (f dt f j (t-T)|x(T)I PdT)1/P G G < (J |x(T)IPdT f j (t-v)dt)l/P G G G~/:1Ip,G The cases p = 1 and p = oo are left as exercises for the reader. (VII 3.iii) If x E L (G), 1 < p < + oo, then J x - x in L (K) as E O0 for any compact K c G; if x E L (G), 1i x uniformly in K. If x E L (E ), then J x - x in Lp(Ea). 17

Proof. Again, let us assume first 1 < p < + oo. Let K be any compact subset of G and let G be any open set with compact closure clG such that o 0 K c G c lG c G. Then x E L(G ). Let 8 = dist (K, bd G ), and take 0 < E < 8. Then y = J x is certainly defined in all of K. Let q > 0 by any arbitrary number. By Lebesgue integration theory we know that there exists a continuous function z(t), t E clG, with (J Ix-zIPdt) 1/p< r1/3. Then z(t) o Go is uniformly continuous in clG and there is some 8' > 0 such that t, t' c 0 clG, It-t'I < 5' implies Jz(t) - z(t')J < (3- jG 1 )l/P. Let us assume 0 < S < min[b,5']. Now, for t E K, we have (J z)(t) - z(t) = f j (t-T)z(T)dT - z(t), G and since the sphere q of center t and radius ~ is completely contained in GO and hence in G, we have fG j (t-T)dT = 1 and finally IJ z(t) - z(t)l = If jE(t-T)[z(T) - Z(t)]dTl G < f j(t-)z() - z(t) d~ <(3-1 1GoI ) 1/p q for all t E K. Finally ( G JI z-zI dt) /< r/3, and since IJ X-XI < IJ (X-Z)) + IX-Z| + IJ Z-ZI, by Minkovsky's inequality and (VII 3.ii), we have hJJ x-xJpdt < JJ x-xJpdt < (f (J (x-z)Idt) 1/p + (f Ix-zIPdt) /P Go Go Go + (fJ J~z-zP dt ) < /3 + /3 + r/3 18

for all E > 0 sufficiently small. The first part of (VII 3.iii) is proved for 1 < p < + o. We leave the cases p = 1 and p = o as an exercise for the reader. To prove convergence almost everywhere in G, we note that, given e Lloc(G), then almost every point t E G has the property that Iql-l1f x() - q x(t) dT + 0 as es O, where q denotes the sphere of center t and radius e, and q c G for all s > 0 sufficiently small. Then IJ (t) - x(t)1 = IfG J~(t-r) x(T)dT - x(t)l < IIq j(t-~)Ix(7) - x(t) Idr < c ~ fqlx(T) - x(t)ldt = ccIql fq| x(T) -x(t) dt, where c is an absolute constant. This proves that IJ (t) - x(t)l * O as -* 0, and this holds a.e. in G. The second part of (VII 3.iii) is immediately proved by taking K so large that, if G' = G-K, then fG, I Pdt < Tip. Then, by force of (VII 3.ii), we have fG, I JgxlPdt < T, and finally (f IJ x-xldt)/ = [(f +f)IJ x-xlPdt] /P G G' K < [2PfI J xlPdt + 2P f IxIPdt + f I Jx-xlPdt]l/P G' G' K < (3P2p)/ p = (2/P3(2). Here we have used the inequality I a+IP < 2p(c? +P) for all a, P real which is immediately proved by taking y = max(Ical,II), and noting that Ix-IP < (2y)P < 2P(IaIP+IoiP). 19

Now let us prove the.third part of (VII 3.iii), which requires a more subtle argument. Let K be a compact subset of G made up of points of continuity for t in G. First x(t) is certainly continuous on K, hence uniformly continuous, and given r > 0 there is some S > 0 such that t, t' E K, t-t'J < 5, implies Ix(t)-x(t')l < q. Moreover, for any t e K, there is some open sphere U(t) of center t such that jx(t)-x(Q)J < n for all t e U(t) n G. We may well assume that each sphere U(t) has closure clU(t) c G and radius < 5/2. Then finitely many of these spheres, say U(ti), i = 1,...,N, cover K, and hence their union is an open set G with compact closure and K C G C clG C G. Let 5" = dist (K, bd G ), and let 5i be the radius of the sphere U(ti), i = 1,...,N. Let us assume 0 < E < min[,/2,",i i = 1,...,N]. Then, for every t E K we have as before I(JEx)(t)-x(t)l = Ij j(t-T)x(T)d - x(t)l G < | j(t-T)jx(T)-x(t)jdT, q where q is the sphere of center t and radius e. For every T E q we have JT-tJ < E < /2. On the other hand T E G, T E U(t ) for some i = 1,...,N, and hence IT-tiI < 5. < /2. Thus It-t. I < It-TJ + jT-tiI < T/2 + E/2 = 2, with both t and ti points of K. Then jx(t)-x(ti)j < q and Ix(T)-x(t)l _< X(T)-X(ti)I + IX(ti)-x(t) I< K + T = 2. We have now I(JEx)(t)-x(t)l < 2q I j (t-T)dT = 2r, q 20

and this relation holds for all t e K and ~ > 0 sufficiently small. The proof of the fourth part of (VII 3.iii) does not present difficulties. Indeed, given -r > 0, we take R > 0 so large that flx(t)lPdt < Tr/2 when the integration is performed in Itl > R/2. Now for 0 < ~ < R/4, we have f jJx-xfPdt f + f > Jx-x'Pdt E tli > 3R/4 Itl < 3R//J < 2p J IxPdt + 2P | IxlPdt Itl > 3R/4 Itl >R/2 + J IJx-xIPdt. t < 3R/4 E The first two integrals in the last member are < r, and the last integral approaches zero as E ~ O by force of the first part of (VII 3.iii) already proved. Remark 1. In the third part of (VII 3.iii) the hypothesis that the compact set K is made up of points of continuity for x(t) in G is clearly stronger than the hypothesis that x be continuous on K. The conclusion would not be true under the latter. Remark 2. We mention here that, if f E L (E ), 1 < p < + oo, and h denotes any vector in E, h = (hl,...,h ), then IIf(t+h)-f(t) Ip - 0 as h - 0. In other words, f If(t+h)-f(t)lPdt + 0 as Ihi 0. E The reader may consult, for instance, E. J. McShane [77q], p. 230, (42.4s). This remark may yield a new proof of the second part of statement (VII 3.iii). 21

We need the following properties of uniformity. (VII 3.iv) If (f) is a family of functions f E L (G), 1 < t < + oo, all zero in E - G, and such that IIf(t+h)-f(t)|| +- 0 as h -+ 0 uniformly with respect to the element f E [f), then IIJ f-f|lp - 0 as E + 0 also uniformly with respect to the element f. Proof. We assume first 1 ~ from clG. We have now V J IJ f(t)-f(t)lPdt = J [J j (t-T)(f(T)-f(t))dT] dt E E E V V V < J [i j (u)lf(t-u)-f(t)ldu]Pdt E E V v f [f (j (u)) /Plf(t-u)-f(t) (j (u)) /qdu]Pdt EE V V < f (j E(u)l f(t-u)-f(t)lPdu)(fjE(u)du)P/qdt E E E V V V The last integral is equal to one. Now, given 1 > O, we can determine a > 0 such that |If(t+h)-f(t)I|p < r for all |hl < a. Now we have, for 0 < e < a, IlJ f-fllp < J j (u)du J |f(t-u)-f(t)|Pdt, P —E E v V where j (u) = 0 for all ul > E. Thus, we may restrict the first integral to the solid ball jul < ~ < a, and then for any lul < ~ < a, we have Ijf(t-u)-f(t)|| K. Thus, 22

IIJ f-f < vn(f j (u)du)/ - E V for all 0 < E < a, and any element f e {f). The analogous proofs for p = 1 and p = o are left to the reader. (VII 3.v) If (f) is a family of functions f E L (G), 1 0, the smooth functions J f(t), t e E, are equicontinuous in E. Proof. Let E > 0 be a fixed number. We assume first 1 0 be any positive number, and let a > 0 be so chosen that -1 v 1/p IIf(t+h)-f(t)ll < ~(K e )/ for all IhI < a and f E [f). We have now, for ihl < cy |J f(t+h)-J f(t)l If j (t+h-T)f(T)dT - J j (t-T)f(T)dTI E E E E = I j~(t-T)[f(T+h)-f(T)]dTj E E IJ j~(u)[f(t+h-u)-f(t-u)]dul V < J (j (u)) /_u)f(t+h -u)-f(tu)l (j (u))l/qdu E < (fj~(u)lf(t+h-u)-f(t-u) I)l/p(Jj (u)du)l/P, Ev E where j (u) = 0 for lul > ~. Thus, we can restrict the first integral to the solid ball lu| < e, where certainly I je(u) < -, and then 23

IJ f(t+h)-JEf(t)l < (Ke )- / (J If(t+h-u)-f(t-u)l' du)/ Ev < (K -v)l/P. (K-16v)1/P for all Ihi < a, and the fixed value of ~. Analogous proof holds for p = 1 and p = oo. (VII 3.vi) If x E L (G), 1 < p < + oo, possesses generalized derivative y = Dx (G) of some order =,...,), then for every compact subset K of G and all 0 < ~ < 5, where 5 = dist [K,jG), we have (D J x)(t) = (J D x)(t), t E K, and J x e Coo(K). Proof. For t e K and O < ~ < 5 we have (D J x)(t) = Dt (t)(T)dT f ( (t-))x()d SG G Here j (t-T) E Cw(G) since j (t-T) = 0 for It-TI > e, and the solid ball It-TI < e is completely in the interior of G. By force of (VII 2.2) we have then lal a DJ x)(t) (-1) f D j (t-~)x(T)dT E G IT E (-1) I j (t-T)(D x(T))dT (J Dx)(t) and this relation holds for t E K and o < E < 5 = dist (K,3G). 24

(VII 3.vii) If x E L (G), 1 < p < + 0, possesses generalized partial derivatives D x E LOC (G) of all orders IaI < m, then for every compact set p K c G. we have i|Da X DaJ xlp K 0 ass -~ 0+ for every taj < m and compact subset K C G. If K is made up of points of continuity for x and all D x in G, 0 < jaI < m, then J x -+ x, D xJ x - D x as ~ - 0 uniformly on K, 0 < lal < m. A corollary of (VII 3-.iii) and (VII 3.vi). (VII 3.viii) If G is connected, if xl(t), x2(t), t E G, are elements of loc L (G) possessing the same generalized first order partial derivatives a.e. in G, that is, Dix1, Dix2 e L (G) and Dixl(t) = D x2(t) a.e. in G, i =,...,v, then xl(t) = x2(t)+c a.e. in G for some constant c. Proof. By (VII 3.vi) we see that for every closed interval I = [a,b] c G and every E > 0 sufficiently small, (JEXl)(t), (J x2)(t) are of class C [a,b] and have the same first order partial derivatives on [a,b]. Thus, (J x1)(t) = (J x2)(t) +c for all t E [a,b]. Since G is connected, the constant c is independent of [a,b]. By (VII 3.iii) we have IiJ X -X, 2+ O IJ1 2XX|| O0 as ~ -* 0. Hence c + c for some constant c as e - O and iX l-X2jlI = = +c 211 i2 1, c meas[a,b], where c is independent of [a,b]. Thus, xl = x2+c a.e. in G. A corollary of (VII 5.viii) is that, if x1, x2 E L (G) possess generalized partial derivatives of all orders a, 0 < JaI < m (all in L (G)), and the derivatives of orders iaj = m coincide a.e. in G, then l-X2 = P is a polynomial in t,..., of order m. 25

(VII 3.ix) If x E L1V (G), 1 < p < + x, and y is continuous in G with compact support K C G, with Lipschitzian partial derivatives of all orders y < m-l (and hence bounded partial derivatives of orders < m in G), then for any a with 0 < acl < m we have Ce jai j a f (D x)y dt= (-1) f x(D y)dt (VII 3.1) G G Proof. Let G be an open set such that K c G C clG C G, and let.. o0 y o o ~ = dist (G,aG) > 0. For 0 < < ~ <, then we have D (J x(t)) = J D x(t) for all t E G. On the other hand relation (VII 3.1) is elementary for x(t) 0 replaced by (J x)(t). Then (D x)y dt= (Df(J~x))y dt (-1)lal I (J x)(D y)dt. Go Go Go N'ow IJs(D x) - D x 0 IJX-XllIIp O where the Lp norms are taken in G By a lmt we obtain l a a passage to the limit we obtain f (D x)y dt = (-1) f x(D y)dt, and G Go either integral is equal to the corresponding integral in (VII 3.1). 26

VII 4. SOBOLEV FUNCTIONS AND SOBOLEV SPACES Wm(G) p For any p and m, 1 < p < _, m = 1,2,..., let us denote by Wm(G) the set of all real-valued functions x(t), t E G, possessing generalized partial derivatives Dx E L (G) of all orders a = (al,...,a ), 0 < < m. For every x E Wm(G) we denote by jIXlI the number p p = xa < m G 1/p (VII 4. 1) z f InP dt As we shall see below Wm(G) is a Banach space with norm Ilxllm. This norm p p |x||m may be indicated also by Ilxl G or norm IIxim in G. p pG p Analogously, for any p and m, we shall denote by Wm (loc, G) the set of a loc all x(t), t E G, possessing generalized partial derivatives Dx E L (G) of all orders 0 < tal < m. If G is any open subset of G with compact closure cl G in GG G c clG C G, then every x E Wm(loc,G) has a restriction in G which is an element of Wm(G ). Functions z E Wm(G) need not be continuous in? o p G. though we shall see in (VII 8) that this is the case if mp > v. Also, we shall prove in (VII 9) certain "fine properties" of functions z E Wm(G) which are mild continuity properties with respect to the single coordinates 1 v F x,.,x. Functions W(G) which are continuous in G are said to be absolutely continuous in the sense of Tcnelli, or ACT in G. As examples of functions z E Wm(G) which are not continuous, take z(x) = xlh, h > 0, for x e G = [Jxl < 1]. Then z E Wnm(G) whenever (h + m)p < v, p > 1, m > 1. For p instance, for v = 2, z(x) = Ix-l/2 belongs to W 1(G); for v = 4, z(x) = Ix-1/2 belongs to W_(G). We shall say that a sequence of functions xk(t), t e G, k = 1,2,..., 27

converges in Wn'(G) toward a function x(t), t E G, if x, xk e Wp(G), and Xk * x, DGXk - Dax in L (G) foralla, 0 < I cJ < m, that is, IIxk - xlI + 0 p p as k -+. We shall say that Wm(G) is a Sobolev space, and that its elements x p are Sobolev functions. From (VII 2.i) and (VII 2.ii) we deduce immediately the statements: For v = 1, if x(t), a < t < b, is AC in any closed internal [a,b] c (a,b), then x E W (loc,(a,b)). In particular, if x(t), a < t < b, is AC in [a,b], then x E W ((a,b)). Here a, b are finite. For v > 1, if x(t), t E G, G c E, G open, is Lipschitzianin G, then V x e W (G) if G is bounded, and x E W (loc,G) if G is unbounded. 1 1 (VII 4.i) Wm(G) is a Banach space with norm J|xJ|. p p Proof. If N is the total number of (distinct) partial derivatives of order Q, 0 < J <i m, then obviously Wm(G) c L (G)x...xL (G) (N times). All we have to prove is that Wm(G) is complete. Indeed, if [xk] is a Cauchy sequence in Wm(G), hence lixk - xhm + 0 as h, k c, where the norm is defined by (VII 4.1), then IIxh - xkli - 0,D xh - DxkxD + 0 as h, k +o in L (G), for every 0 < Jal < m. Since the space L (G) is known to be comP P plete, there are elements x E L (G), ya C LP(G), such that xk - x and D Xk+ Ya in L (G), for 0 < lal < m. All we have to prove is that yG = D X, o_ iI <l L

Now xk * x in L (G) certainly implies JG Xk Df dt t G X Dcp dt for every fixed cp e C (G) and any a < m Similarly, Dk y in L (G) certainly implies Gf (D X)cp + Ga cp dt. Since x E W'(G) we have f (D xk) p dt = (-1)I lG xk (D p) dt, k = 1,2,..., Ia <I m, and hence, as k - o, we have also G Ya (p dt = (-1) IIG x(D p) dt, I a < m. That is, y = D x, 0 < I al < m. We have proved that x e W'(G) and that Wi(G) is complete. p (VII 4.ii) (Leibnitz rule in Sobolev spaces). If x WIn(G) and y e Wq(G), P q 1/p + 1/q < 1, and 1 < X < + cc is so chosen that l/p + l/q = 1/h, then the P a.e. in G. product xy e W(G) and Da(XY) = E( ) Dx D y a.e. in G. A corollary of (VII 3.viii). (VII 4.iii) If x e WIm(G), y = Da z -= y, 0 < al i m, 0 < I< I < m, 0 < la + I <i m, then z = D x a.e. in G. A corollary of (VII 2.ix). The following criterion is often used. (VII 4.iv) Let y (t), t E G, 0 < Ia1 < m, be given functions y E L (G) 1 < p < +, let Rk, k = 1,2,..., be a sequence of open subsets of G with Rc Rk+l, Rk G as k, and let x(t) t Rk, k = 1,2,..., be a sequence of 29

functions xk E Wp(R ) such that IY - D xk R 0 as k 0 < ac < m. px pThen y = y is an element of WpT(G), and y = D y a.e. in G, 0 < Ial < m. A corollary of (VII 2.x). Let us consider now the class L of all functions x(t), t E G, which are Lipschitzian in G together with all their partial derivatives D x, 0 < jai < m - 1, and hence possess bounded partial derivatives of order m a.e. in G. Then L is certainly a normed space with the norm IIxiim defined by (VII 4.1). We shall denote by H?(G) the completion of L with respect to the p norm IxiII~. We shall prove in (VII 10), under mild conditions on G, the basic identity wm(G) = (G). (VII 4. 4) p p All we can deduce from (VII 4.1) is that HI(G) c Wm(G) and this is true for P P every open set G c E. Indeed, L c Wm(G); hence, the completion Hi(G) of V p p L with respect to the norm 1 || M, being the smallest complete set containing L, certainly is contained in W1(G), or?F(G) c Wm(G). Analogously, we may consider only the class L of all those elements x of L having compact support K c G. We shall then denote by Hm the' cmmx op pletion of L with respect to the norm Ixilm. We shall prove in (VII 10), 0 p under the same mild restrictions on G, that HI (G) is the subset of all those op functions x E W'(G) "whichare zero on the boundary" aG together with all their partial derivatives of orders 0 I Gai <m - 1, according to the definition of boundary values we shall discuss in VII 7, 8. 30

Relation (VII 4.4) is easy to prove for G a half plane, say the half + 1 1 plane E of all t = (t,..,tV) with t > 0. All we have to prove is the following statement: (VII 4.v) If x e Wm(E+), then there is a sequence xk, k = 1,2,..., of pV + m00 functions xk Wm(E ) n C (E ) such that lIxk - XIIm 0 as k. pV V k p To prove this statement we first denote by h = (h, 0,... 0) a vector with h > 0, and we note that, if y(t) = x(t + h), then y E Wp(F) where F is the half space t- > -h~ hence, also y E Wm(E). By Remark 2 in (VII 3) we know that IlY - Xl| E+, JID Y - D xj + + 0 as h -+ 0 +, < c < m. V V Thus, it is enough to prove our statement for y. Now we take a function E e Cw(E ) with the following properties *(t) = 4(t ) - c < t < t 00, V?(tl) = 0 for tI < -3hl/4, i(tl) = 1 for t > _-hl/4, 0 < < 1 otherwise. We define now z by taking z(t) = 0 for t < - h z(t) =y(t) (t) for t > -h Now z is defined in E and we have to prove that z has generalized derivative V D z in E given by Y = O for t < -h Y = y D for t > -h 0 < al < m. Indeed, if p e C (E ), then JE Y t = IF E ( )(~iy)(D 1-Pr)cp dt, - IY'<M Inei ECo v 1Ev3 where all functions (D 4r)(P are actually in C (F). Then 0 JE Ya dt = [FY (-l)t' (a)'D((-Da )cp] dt [= FY ~ ~ (-1) l (1) Dcp Da- dt where ~ ranges over O <1B <a and ~ over O < y <1B. For y = c we have y = D = a and the corresponding term in bracket is (-1)I I (D or). All other 31

terms in bracket have sum zero, because of the identity E (-1) l()( ) = O. Thus fE Y p dt = (- 1)I IE (Z )(D p)dt, V V for all cp c C (E ), and we have proved that Y = C in E, 0 < jai < m. By O VI force of (VII 3. iii) (fourth part) we conclude that IID (J) - DZpE as E + 0 +, 0 < jal < m. In particular, we have lD (J z) z DZl E+ p+ as E - O. We can now easily choose hk and sk sufficiently small, and take Xk= JEkZ VII 5. EMBEDDING THEOREMS IN W (E ) p v The Sobolev embedding theorems are particularly easy to state and prove for Sobolev functions in a half space. We shall introduce below (VII 9) the concept of regions G of class K in E, and then the embedding theorems V will be translated immediately in terms of Sobolev functions x e W1(G) in a region G of class K. In the T-space E, t = (t,...,tV), we shall denote by E the part of E V V V 1 + with t > O. Then E is an open subset of E V V We shall often use polar coordinates in E, say (r, w), r = Itl > O, ( = (,**.,)v) E S, where S is the unit sphere jtj = 1. We denote by S+ the half unit sphere [tI(t) = 1, t > 0], by dw the usual area measure on S modified by a constant factor so that AS + dw = 1. We have then, for every x E L1(E ), 32

fE+ x(t) dt= f0 fS+ x r drdw. (VII 5.i) Let x(t), t E E, be an arbitrary function x E L (E ), 1 O, Itl < R], and possessing generalized first order partial derivatives D x, j = l,...,v, in E of class L (E ). Then, there are functions h (t), + 00 E+ E+ t E E, hj C (E ), h EL (E ) for every 1 < s < v/(v-1), h of compact V 3 V s v - j support [tltl > 01 Itl < 4 R], such that x(t) = - f (D.x)() h.(-t) dt a.e. (VII 5.1) j=l F(t) and where F = F (t) denotes the part of the T-space where T > t. For V V v = 1 we have h = 1 for O < T < 4 R and (VII 5.1) reduces to x(t) = - t Dx(T) b(T) dT = - f Dx() dT. Proof. Let b(r), 0 < r < + 00, be a function of class Cw on 0 < r < + c0, equal to one on [0, 5R] and zero on [6R, + ao)]. Let cp (t), t E E, be any function (P E C (E+) with compact support x contained in [tft > O, Itl < 2R]. O v It is convenient to extend (P to the whole t-space by taking cp = 0 in E -E, V V so that c so extended is now in Cw(E ). For every t e E with Itl < 2R and 0 V V X e S we certainly have (T - rw) = 0 for r > 4R; hence fo (6/6r)(p(T-rC') b(r)dr = of (6/r)(P(t-rm)dr = @p(t). + 00 4R Then, by integration on S we have also 35

Cp() = f +dt f0 a cP(-rw) b(r)dr S + 00 r = f +d o zE. (Dj p)(t-rw) w.b(r)dr S 0+ 00 -v+l vv 1 ++ J (Djcp)(T-rw) (w. b(r)r ) r vdr j=1l + J =v f+ (Dj p)(t-T)h (T) dT, -v+l + where h.j() = C b(r)r, T E, j=l,...,v. As we shall see below h. E L (E+). By Fubini's theorem we have now J 1 v I += |+x(t)W(t)dT = Z J+ x(t) dT f+ (DjCp)(t-T)hj(T)dT E E E V V V = z Ef hj(T)dT f+ x(t)(Djcp)(t-~)d~. (VII 5. ) v'E E E V V Note that cp has compact support certainly contained in a slab 0 < a < t 0, and the function cp(t-~), as a function of t, has compact support 1 1 1 1 contained in the slab defined by a < t - T < b, or a + T < t O. Thus, Cp(t-T) as a function of t, has compact support in + E, and (VII 5.1) yields V I f hj(t)dt f+ D.x(t)cp(t-~)dt E E V V a E+ f + f hj () Djx(t)cp(t-T)dldt =E E v v and the latter is a double integral which can be restricted to the part of E x E where t - T > (andofcourse > O, > 0). By writing V V

t - ~ = u, t = v, then the same double integral is transformed into I = -v1 fJf h (v - u) Djx(v)y(u)dudv, j=l j J where now the integration is performed in the part of the uv-space where 1 1 1 1 u > 0, O v > - u > 0. By Fubini's theorem we have now = = - D.x(v)h (v-u)dv cp(u)du. uEj 1 E, vl>ul j By replacing v by T and u by t, then we have t E E, T E E, T > t. If for V - any t E E we denote by F = F (t) the part of the T-space where T > t V v V then by Fubini's theorem we have f+ x(t)cp(t)dt =1 f Djx(T)hj (T - t)dT cp(t)dt. E E F This relation holds for every cp E C (E ) with compact support in [tlt > O, Itl < 2R]. By force of (VII 2. iv) we conclude that x(t) = -v1 f+ Djx(T)hj (T-t)dT v at least for almost all t with t > O, ItI < 2R. It remains to prove that hj E L (E ) for every s with 1 < s < v/(v-l). V Indeed o -v+l S v-I f+ Ihj(t) dt = I J I (oj. b(r)r I r ldr &d E S o 0

Note that in the integral (VII 5.1) we may assume T > 0, I T < 2R, T - t > 0, IT - tl < 2R. In any case, 0 < r = I~ - tl < 4R. (VII 5.ii) If 1 O, Itl < R], then x E L (E ) for every q, _- -' q v 1 < q < +, with l/q > 1/p - l/v, and there is a constant K depending only on R, p, q such that ixll v, q = -a, x E L (EB) and ess sup Ixl < K zE1 IIDjXl (VII 5.4) In other words, if W (E+) denotes the set of all elements x E W(E ) with p,R v p v compact support in the solid half ball [tlt > O, Itl < R], then the identity transformation carrying an element x C W (E ) into the same function x as an element of L (EB) is a bounded map W (EB) L (E ). We shall see in q v p,R v q v (VII 5.v) that the same map is also compact (for p > 1, and even for p = 1 under restrictions). Proof. First, let us assume 1 0. Also, note that p can replace q in the relation 1 /q > 1/p - l/v hence, we can assume p < q. Then, for r = It-Tl, we have Ihjl < Kr-v+ = Kr- for some constant K, and then, by (VII 5.1), we also have 36

(vII 5.5) where Djx = (D.x)(T), r = It-Tj, and the integration is performed in the solid ball Irl < 2R. If we take \ = l/q, k2 = l/p - l/q, X3 = l/p', we have 1' 2 >0, ~ + k2 + k3 = 1. By Holder inequality for three factors we have then jx(t)j <K ~.1 [ID xIP r ] l/ ID.xPdT] j-1 IEP D x p r [r-V+p d, (VII 5.6) where again the integrals are taken in Irl < 2R. Since -v + Ep' > -v the third factors are below a fixed constant. The second factors are also l-p/q' finite and equal to IIDxl P/q By taking powers q in (VII 5.6), by integration in G, and interchanging the order of integration, we have f Ix(t)lqdt < K1 V_ IDx |qq f IDxPdT r v+Eqd r<-R j<R D x 1p r<R r<R Since -v+Eq > -v the last integral is below a fixed number. By Torelli's theorem then Ix(t)1q is L-integrable in G, and by Fubrini's theorem the transformation above are valid. We obtain now f Ix(t)q dt < K" ZE 1 IID x q r<2 for some constant K", and (D5.4) follows. If v < p, q = o, then we take 2s = 1 - v/p > 0, and instead of (VII 5.5) we write 37

Ix(t)l < K Z v1 fID xlr2p-V/p'dr and by Holder inequality x(t)l < K Z j=1 Dx Pd r-v + 2Ep' The argument is now similar to the one above. Let us consider the case p = 1. Again we take x = v-1. Finally, the relation 2E = 1 + v/q - v/p > 0 reduces to 2s = 1 + v/q - v. and jx(t) < K 1 | (D X Djl/q r2V/q) D x / dT If we take X! = l/q, = l-l/q, we have 1 > 0 2 > O, + 2,, by Holder inequality, Ix(t)| < K Zv fO D xi -v +2x3 I/ 1D 3 As before, by taking powers q, by integration in r < R, and interchanging the order of integrations, we have f Ix(t)lq dt < K1 Ev D l JID.fdT r-v+2 qdT, r<R j= j 1 r<R r<R where -v+2Eq > -v. The last integral therefore, is below a fixed number, and the estimate above shows, by force of Tcnelli's and Fubini's theorems, that Ix(t)lI is L-integrable in E. Finally, ~v~~~ ~v I jx()JI dt < K" Eiv ID q1 and (VII 5.2) follows for p = 1. 38

(VII 5.iii) If v > 1,1 1, if x e WI(E+) with compact support pv contained in the solid half ball [tItl > 0, Itl < R], then x E L (E ) for q V every 3 <q < + oo with l/q > 1/p - m/v, and there is a constant K depending only on R, p, v, m, q such that x|lli < K _ liD X|. Also, every genq - l =m p eralized partial derivative D x of order 0 < f| < m - l is of class L (E) for every 1 < q < + Xo with l/q > 1/p - (m-0l )/v, and there is a constant K depending only on R, p, v, m, q, a such that ||D x|| < K Z D~x||, 0 < jai < m - 1. II~aXII q~ZPi orjcx <m-l. (VII 5. 7) In other words, we have here bounded maps W R(E*) + L (E ) as mentioned p,R v v after statement (VII 5.ii), and, as we shall see further, these maps are also compact (for p > 1 and even for p = 1 under restrictions). Proof. If = l/q - l/p - m/v, let P = p' P,...Pr- q be the numbers defined by /Ps+l= /s -l/v +E/m, s = o,l,..., m - 1. Then, by (VII 5. ii) we conclude that all derivatives Dax with jal = m - 1 are in L, all derivatives D x with IaC = m - 2 are in L, and so on. Thus x Pl P2 is in L, or x E L, as stated. The corresponding statement holds for the Pm q derivatives since S > 0 above can be any positive number. The remaining part is a corollary of (VII 5.ii). (VII 5.iv) If 1 1, m > 1, mp > v, if x E W (E+) has compact support contained in half solid ball [tit1 > O, Itl < R], then x and all 39

partial derivatives D x of orders 0 < Ita < m - v/p are continuous in the closure of E. Also, there are constants K, depending only on R, p, q, m, v, a, such that a + D x(t)| < K IZ'i M|x|lpt, O < lat < m - v/p, t c cl E. (vII 5.8) Also, there is a function x(h) > O, h = (hl,...,hv) E E, depending only on + + R, p, q, m, v, a, such that, for all t e E, t + h E, we have V V a a JD x(t + h) - D x(t)| < K X(h) ZlI =E m lxlp (VII 5.9) This is Sobolev's imbedding theorem for Wm(E+). It will be translated p v in terms of arbitrary regions G c E (of class K) in (VII 11). This statement shows that, not only x and all partial derivatives Dax with 0 < fa| < m-v/p are continuous on the closure of E, but also that the identity transformation carrying an element x E Wm (E+) into the same function x, or into its derivapR v tives D x, 0 < Kau < m-v/p, as an element of C, is a bounded map Wm (E+) * pR v C(clE ). Moreover, the same map is compact. Namely, the functions x E v W (E ) with tlxl m< M for some constant M, certainly are continuous and equipR v p+ continuous functions on cl E (and so are their derivatives D x, 0 < tat < V -- m-v/p. Proof of (VII 5.iv) For any multiindex a = (a,... a ) with 0 < atl < m-v/p. 0 < laI < m - 1, let g = Ifx. Then the v first order partial derivatives D'g, j = 1,...,v, are derivatives of order tal + 1 m of x. If tal K m - 2. then D.g C L (E ) j = l,...,v, for every 1 < q < + co with l/q > l/p - (m - lal - 1)/v. If l/q' + l/q = 1, and we take q > p with l/q > l/p - 40

(m - 1a - l)/v, then l/q' = 1 - l/q < 1 - l/p + (m - la! - 1)/v = (l/v)(m - ca - v/p) + (v- 1)/v, where m - [ - v/p > O. Thus, if we take q > p with l/q larger than and sufficiently close to 1/p - (m - a I - l)/v, then l/q' > (v - l)/v, and 1 < q' < (v - l)/v. If ICa = m - 1, then laI + 1 = m, and Djg E Lq(E ) for q = p, j = 1,..., Again, for l/q' + l/q = 1, then l/q' = 1 - l/q = 1 - 1/p = 1 - 1/p + (m - CX + 1)/v = (l/v)(m - lal - v/p) + (v - l)/v, and again 1/q' > (v - l)/v, and 1 < q' < v/(v - 1). In any case we have determined q, q' > 1, with l/q + 1/q' with 1 < q' < v/(v - 1), and this shows that q > 1. The functions hj are known to be in L (E+) for every 1 < s < v/(v - 1), hence hj e L,(E+), and on the other hand Djg E L (E ), for the chosen j q 3 q v q > p and l/q' + l/q = 1. By (VII 5.1) we have now, a.e. in E (t) J =1 J( t ) D g(T ) h (T-t)dT, F(t) J and by Holder inequality also ig~t)'<Zv /fF+ J ~) i/q )1/q' Ig(t)l = Fzy f ID.g(-) Iq d1C (_rt) q d'' V V K j=1 IDj gll By force of (VII 5. iii) we have then IDax(t) I = I g(t) < K1 = mlD X<Kp' This proves relations (VII 5.7). +,l + FortEE andeveryh =(h,...,h)eE withtheEwehavenow Vw,

g(t + h) - g(t) = -.=b F D.g(T)h (T-t-h)dT - f D.g(T)h.(T-t)dT, -v(t+h) F (t) F + (t) T 1t+h) V F(t) = EIT e E T > t F (t+h) = liT E T.> tl+h Let tl min [tl, tl+ h, t]=max [t t + h ] and let H1 H2 be the sets + -1 1 =1 + =1 1 H = [tit E E t < t < t ] H2 = [tt E t < t ]. Then Ig(t + h)-g(t) <- I f IH1 Dj g(T)h J(-h)d-I + V 1fH Djg(T) [j(T-t-h) - hj(T-t~ dT = v1 (J + J ). j=1 jj=2 We have now I J. 1 d< J+I Dg(T) lq dT ( fl h (T-t-h) - h.(T-t) ~ dT) 11 V V < lDj.gl Jh (u-h) - hj(u)lJ, = IDJglgI X (h). By Remark 2 in (VII 4) we know that Xi(h) + 0 as |hi + 0, j = 1...v To estimate Jjl we may well observe that we can replace the domain of integration H1 by the subset H of all points t c Hwith It| < 2R. Then H < 2vl Rv-1 (tl-i ) < 2-1 R-1h. Also, we shall take numbers s' and X so that q' < s' < v/(v-l), % > l, 1/\ = l/q' - l/s'. Then s' > 1, q' > 1,' > 1, 1/k + 1/q + l/s' = 1, and by H6lder inequality we have J < ( 1 dT) fH ID g(T)qdT) (JH h(T-h) dT < 1/< K(2V <K IIDjg[q I Ho_/ < Djoj.l Ihi 42

These estimates for Jj and J., together with (VII 5.iii) yield (VII 31 J2 5. 8). Statement (VII 5. iv) is thereby proved. Remark 1. Statement (VII 5. iv) is certainly not valid without the assumption mp > v. For instance, for v = 1, m = 1. p = 1 the AC functions xk(t), t c I = (0,1], defined by xk(t) =1 - kt for 0 < t < k-, xk(t) 0 -l 1 -.1 li=t h= ).... for k <t < 1, are all in W1(I) with |lxil = (2k)- x'! = 1, k = 1,2 Clearly, they are not equicontinuous on [0,1]. (VII 5.v) If v > 1, m> 1, p > 1, if x E Wm(E+) is any function with compact _ - p v support contained in the half solid ball [tltl > 0, Iti < R], then there are functions X(h) > 0 h = (h... hV) E E with X(h) + 0 as Ihl + 0 depending only on R v, m, p, q, such that j-D x(tch) - D x(t)Vj < X(h) l j]Yx, 0< la < m - 1 (VII 5.10) provided, 1 < q < + oo, l/q > l/p - (m-lal)/v and either p > 1 or p > 1 and a < m - 2. For p = 1,!a = m - 1, let D (a) a > 0, Iol = m denote the supremum of fX IEx(T)ldT for all measurable subsets X of E with Ixi < a. Then there are functions x(h) > 0 as above and constants K1, K2 depending only on R. q v, m, such that D\ x(t+h) - Dx(t)| < X(h) _Ill Dxl + K1 (K hi) (vII 5. 11) provided 1 < q < + 7, l/q > l/p - (m-fat )/v = 1 - l/v. 43

Proof. We assume first p > 1, m = 1, a = O, l/q > 1/p - 1/v and also h = (h,... hV) with hl> 0. Take l/p' + 1/p = 1, 1 = v/(v-l) and note that l/q + 1/p' - 1/r = 1/q - 1/p + 1/v > O. If we define ~ by taking 6(1/q + 1/p') = r(1/q + l/p' - 1/r). then we see that 0 < e < ~, and that (q - E)/q + (r - ~)/p' = 1. From (VII 5.1) we have x(t+h) - x(t) = - j1 [. D.x(T) h.(z-t-h)dT - f D.x(T)h (T-t)dT], j1 - + j F (t+h) F (t) V V F (t) = [TIT E E, T > t ], F(t+h) = [TIT E E, T > t + h]. V V V V T Let l = min [t, t + h], t = max [t, t + h ], and let H1, H2 be the sets + -1 1 =1 + =1 1 H = [tit E E, t < t t ], H2 = [tit E, t < t]. Then 1 vV 2 Ix(t+hl-x(t)l < v =11 rH Dix(Tr) hj(T-h)dT| j=l+ fI H D x(T) [h.(T-t-h) - hj(T-t)]dTI= (Jjl(t)+J (t)). 2 J In J the range of integration H1 can be restricted to the set H of all ji_1 0 points t E H with ItI < R]: hence, I H < KIlhl < Klh| for some constant K. Note that 1/q > 1/p - l/v, that we can well assume 1 i/p - 1/v must hold for some number p, 1 0, and IJ1(t) I < IH D xI r -/ D x (r ) dT

If we take 1 = l/q, k2 = 1/p - l/q, k3 = l/p", we have >i, X2, >3 > 0, k1 + A2 + \3 = 1, and by Ho6lder inequality also IJj1(t)I < IDjxIP r VqdT l / D.xI Pd3 f r-v+EP"l d r dp j'1/p where |tl < R. Since -v +Ep" > -v, the third factor is below a fixed constant. By taking powers q, integrating with respect to t in Itl < R, and noting that then r = It-Ti varies in [o,2R], we have f Jj (t)lqdt < K1 ID- x dr| t<R j f ij Since -v +Eq > -v, the last integral is below a fixed number, and thus IJjl(t)i dt < K" f IDjxI |dr t<R H Again, by Holder inequality with exponents i1 = = 1-/p, 1, A2 > 0, we have f |Jjl(t)l |dt < K1 IH i q/p - lID qIPd anc finally, since H _I < Kihi, also ( I |Jjl(t) l qd t < K Dj h/P. We shall now consider Ji2 We have

IJj2(t)l < J IDjx(-)I Ihj(r-t-h) -hj(T-t)ldT H2 < J (IDjx(J)lp/q Ihj(_-t-h)- hj(v-t)l(B-)/q H2 (ID jx()Ip(1/p - 1/q) (ih (_-t-h)- hj(-_t)l(q-~)/p')d. If we take 1 = l/q, 2 = 1/p-l/q, X3 = 1/p', we have >kl, X2, 3A > 0, 1i + 2 + k3 = 1, and by Holder inequality with three factors, also IJj2(t) _ < IDjx(T)lp Ihi(T-t-h)- hi(_-t)l -d~ 1/q D. x()x( I Pd-) p -/q I |h (,-t-h)- h.(-t)I-E di VP The second factor is equal to X(II P)/q Since j = v/(v-l), and hj E L for every s < B, we see that the last factor is finite. If we take T-t = u, the last factor is < |1hj(u-h)- hj(u) ||(-)/. Thus, by taking powers q and integration, we have, for some constant K, f IJj2 (t)lqdt < K2 I|DjxllP I|h(u-h)-h(u)II() ItI <R j2 2 I+lDjx(T)lp dT Ihj(T-t-h)-hj(T-t)l -Edt. Ev Ev If we take u = T-t in the last factor we have f iJj (t)Iqdt < K'2 jD jx1 Ihj(u-h)- hj(u) (/P +1) Itl <R j2 2 Since I = v/(v-l), and hj E L for all s < T, we see that the last factor is a function Xj(h) with Xj(h) O0 as Ihl * O, and Xj depends only on R, p, v, q. 46

By taking X(h) = max[Khl th1/P I/p + K- Xj(h)] we have relation (VII 5.9), and statement (VII 5.vi) has been proved for m = 1, p > 1. Let us assume now m = 1, a = 0, p = 1. Hence, q is any number such that i/q > l-1/v, or 1 = p < q < v/(v-1). Concerning J., we have the estimate IJjl(t) _< f Dj.X(T)J r-V+ldT Ho = f (IDjx(T)l1/q r-V+l )(ID.x(T) 1-1/q)d Ho _~ [H"oDj(-)' r(-v+l)qd]/q [H1o'Djx(7)'d1-1/q By taking powers q and integrating on [Itt < R, t' > 0], we have | -(t)lqdt < rJ x(T)|dt]l jDx(T)jdT r(-v+l)qdr. It| <R o j L r < 2R Since q < v/(v-1), hence (-v+l)q > -v, the last integral is below a fixed constant, and f <R j (t)| dt < K1 f IDjx(T)'dTlq Itl <R jo < K1 [0j(IHo )]q (I J<R J(t)'dt)l <' K j(Klhl). Concerning Jj2 we have the estimate 47

IJj2(t)l f I Djx(T)l Ihj(-t-h)-hj(T-t)ldT H2 < I (IDjx(T) l/q Ihj(T-t-h))-j(T-t)l)(IDjx(T~)l -1//) d H2 < f DDjx(T)'hj(T-t-h) - hj(T-t)Iqd 1/q 2H2 and, by taking powers q and integration, also tl RIJj2(t)l dt < rH IDjx(T)Id q-1 Itl <R H2 J ID x(T)IdT) I hj(-t-h) - hj(-t)qdt Since h. E L (EK) for v = v/(v-1), and 1 < q < v/(v-l), we see that hj E j TV v L (E ) and hence X (h)= IIhj(u-h)- h (u)I -* 0 as Ihi - O. Thus, qV j j q t I <R j2 From these estimates for Jjl and J2 we immediately obtain (VII 5.10) for jl j2 m = 1, a = 0, p = 1. So far we have proved (VII 5.v) for m = 1 and p > 1. Let us assume m > 1 and 0 < lal < m-1. Let g = D x, and note that the first order partial derivatives Djg, j = l,...,v, are derivatives of order al +1 of x. If mat = m-l, then al+1 = m, and we can apply (VII 5.v) to g E W (E+). Then either (VII 5.9) or (VII 5.10) hold according as p > 1 or p = 1. If Iai < m-2, then Djg E L-(E ) for every p with 1 l/p - (m-i ao-l)/v, and we can take p to be > 1 and as close to l/p - (m-Jai-l)/v as we want. We now apply (VII 5.v) to g _(E ) with > 1. pv 48

1lg(t+h)- g(t)JlI X(h) Z_1 Dj g for every q with l/q > 1/p - 1/v. Thus, for every q with l/q > 1/p - (m-Ial)/v and by the use of (VII 5.9) and (VII 5.iii) we have ID x(t+h) - D x(t)ll < X(h) +1 ID7xlI< X(h) K Zl1 =m IID'XIp. Statement (VII 5.v) is thereby proved. Remark 2. Relation (VII 5.10) of (VII 5.v) is not valid for Jau = m-l, p = 1, as the following example shows. In other words, the exceptional case p = 1, IcI = m-l, for which we have proved (VII 5.10) instead of (VII 5.9) cannot be disregarded. In the example we consider below we have m = 1, p = 1, a = 0, v = 1, and hence we can take for q any number q > 1. We shall denote by xk(t), 0 < t < 1, k = 1,2,..., the usual piece-wise linear functions which converge uniformly to the ternary Cantor function x(t), 0 < t < 1, on the interval [0,1]. Namely, if Ill, I21, I22, I31, I32, I33, I34,..., Ikl' Ik2'.,Ik2 k-l are the intervals of constancy of x of lengths 1/3, k 1/3.,1/3, respectively, let xk(t) = x(t) for t in these intervals, and let xk(t) vary linearly in the 2k complementary intervals Jkl' Jk2'...,Jk2k. = 2k 2k Then xk has a variation jkl/2 on each interval Jk' s =l,..,2, all of length k Then each k = 1,2,..., is absolutely continlength 1/5. Then each xk(t), 0 < t < 1, k = 1,2,..., is absolutely continuous and nondecreasing in [0,1] with Xk(o) = 0, xk(l) = l, hence llXk|l l< l, Ixkll = Ix(1)-x(o)j = 1, for all k. If we define each xk(t) for t < 0 by 49

taking xk(t) = O, then each xk is of class W (E) where E is the half interval (-00, l). For any given h, O < jhJ < 3, let k denote any integer with 1/3k < Jhj. Then the displacement operation t + t+h takes each point t E Jk = [a, ] into a point t+h either t+h > P or t+h < a. Thus, each t of the interval k+l J' concentric to Jks and length (1/53)Jk s = 1/3 is mapped into an interval either at the right of P or at the left of a. In any case we have Ixk(t+h) - xk(t) (l/)jk = (1/3)(1/2 k) for all t e J' with IJ' I = /3k+ Thus, for any k with 1/3 < Ihl we certainly have ilxk(t+h) - Xk(t)I|q > (2k.3-k-l(3-12-k)q)l/q = 3-1 3-(k+l)/q 2(1-1/q)k Let us prove that it is not possible that Ilxk(t+h) - xk(t)llq < X(h), for all k with 31k < hlI, for some function x(h), 0 < Ih < 3 -2, with x(h) > O, x(h) - 0 as IJh + O. It is enough to prove that it is not possible that 3-1 3-(k+l)/q 2(1-l/q)k < X(h) (VII 5.12) for all k with 3 -< JhJ. Indeed for the minimum k for which this relation -k -k+l holds, we have 3-k< JIh < 3, hence k log 3 > -log JhJ > (k-l) log 3, (VII 5.13) 50

while (VII 5.12) can be written in the form A = -log 3 - ((k+l)/q)log 3 + ((q-l)/q)k log 2 < log x(h) = B. In view of (VII 5.13) we have A = - log 3 - (l/q) log 3 - (l/q)(log 3)((log. 3)- log J!h) + ((q-l)/q) log 2 (log 3 - log Ihi) (- log 3 - (l/q) log 3 + ((q-l)/q log 6) - log!h!, and the last expression is certainly positive for all Ihl > 0 sufficiently small. On the other hand x(h) > 0, X(h) + 0 as Jhl -+ 0, hence B = log X(h) must be negative for all jhI > 0 sufficiently small. Thus, A > O0 B < 0, A < B, a contradiction. We have proved that relation (VII 5.10) does not hold for the elements xk e W (E). The following variant of (VII 5.i) is relevant. (VII 5.vi) Let x(t), t E E, be an arbitrary function x e L (E), 1 O, Jtl < R], and possessing generalized partial derivatives D x in E of all orders lal < m V all of class L (Ev). Then, there are functions h (t),t E E for every = m, h E C (E+), h of compact support contained in [tit' > O, lt I< 4R] such that 01 V C( — x(t) = flFl=m IF(t) (D x)(T)h (T-t)dT a.e. (VII 5.14) where F = F (t) denotes the part of the T-space with T > t. In addition V V h _( + a LS(Ev) for every 1 < s < v/(v-m) if v > m, and ha E C (cl Ev) if v < m. 51

Proof. We use the same notations as in the proof of (VII 5.i). Then for + + cp as assigned in that proof, t E E, rt[ < 2R, and w E S we have m- r ((m/r m) cp (t-rwo)) b(r)dr = ((m-2) )-i m-2 J -m/ m-1) p (t- rdr = = 4R( /cr) c (t-re)dr = cp(t). As in the proof of (VII 5.i) we have now -1) ic j+ ElJf+m (Dv -)(t-ri)(cwb(r)r-v+m v-d cp(t) = ((m-1)!) = E E S V l =m + (D C )(t-TC) h (T)dT, E V where h (T) = wcb(r) r +m E + (11 ( ) Uv The proof proceeds now exactly as for (VII 5.i). It remains to prove that h E L (E*) for the stated s. We assume v > m. We have U s v f+ Jh (t) s ds = J+ Jo 1ow b(r) r T r dr &d E S V <K f6R r-vs+ms+v-1 d -- 0 and -vs + ms + v - 1 > -1 reduces to the assigned inequality i < s < v/(v-m). We are now in a position to prove the following useful variant of statement (VII 5.ii): 52

(VII 5.vii) If x C W (E+) has compact support contained in the half solid P V ball [tltl > O, It < R], if A is any hyperplane of dimension a in E, 1 < a < v, and G c A n clE is an open set in A contained in [tit' > 0, --'- a v a - ItJ < R], then the restriction x* of x on G belongs to Lq*(G ) and 11IIXIL Ga) < K ZlaJ. JID'XIIl (VII 5.15) where K is constant depending only on R, p, a, A, m, q*, provided v > mp, a > v-mp, and q* < ap/(v-mp). If v < mp, then x is continuous on cl E, and V so is x* on cl G In other words, the identity transformation carrying an element x E m R(+ + Wp (Ev) into its restriction x* in G c A is a bounded map (E ) L (G ). pR v Cr a p,R V q a The same map is also compact with a few restrictions as we shall mention below. Proof. We may replace G be a region, say still G, well contained in the solid ball It! < 2R. Let us assume first 1 0. Also note that p < ap/(v-mp), and therefore we can certainly take p < q* < ap/(v-mp). Then, for r = It-TJ we have jh < K r = K r for some constant K and then, by (VII 5.14) we also have r ( i'Nr 7) )(rN (VII 5.16) 55

where D x = (D x)(T), r = [t —T, and the integration is performed in the solid ball Irl < 4R. If we take XI = l/q*, 2 = 1/p - l/q*, >3 = 1/p', we have IX + 1 2 + +3 = 1, \l, X2, X > O. By Holder inequality for three factors we have then Jx(t)l < K Zia:m [ffD xP r a+q* dTl/q* [f CID'xlP d]]l/p - l/q* [I r-v+ dP ]l/p where again the integrals are taken in Irl < 4R. Since -v+~ p' > -v, the third factors are below a fixed constant. The second factors are also finite and equal to jJDX| 1-p/q*. By taking powers q*, integrating on G, and interp changing the order of integration, we have ~ x <' V JDCxllq*-p f IDCxIp d f r cy+q*dT. I x( t) lq dt < K 7,1cxl=m IID XIIID I aG IceI=m p r<hR G Since -a+sq* > -a, the last integral is below a fixed constant. By Tonelli's theorem the multiple integral in dtdT above exists, and by Fubini's theorem the change of order of integration performed above is valid. We obtain now G Ix(t) i=m* for some constant K", and (VII 5.15) follows. The remaining cases, in particular the case p = 1, can now be treated analogously. A theorem analogous to (VII 5.v) holds here, too, and guarantees that the map WpR(EV) + Lq.(G ) is compact. A relation analogous to (VII 5.10) must pR V q* a ~ ~ 5

be proved, and we leave the proof to the reader. Again the case m = 1, p = 1 is exceptional, and for this case the reader will be able to prove a relation analogous to (VII 5.11). The case m = 1, p = 1 is actually exceptional as we can see from the following example with v = 2, m = 1, p = 1. Let cp(u), -c < u < +oc, p(O) = 1, be a function of class CO(-Oo9o), and let zk(t, u), (t, u) E E E [- < t < 1, 0 - o < u < + o] be the functions defined by taking zk(t, u) = xk(t) cp(u) for 0 < t < 1, -ao < u < +oo, and zk(t, u) = 0 for -o < t < O, where xk(t), 0 < t < 1, are the functions defined in Remark 2. Then zk E W (E) n C(clE), zklll _< M, II~z/)tII M, faz k /<UlU I < M for some constant M, and actually Zk E W (E) for a suitable R. On the other hand, the restriction z* of z k l1R k k on the hyperplane u = 0, is the function of x k(t) for which no relation analogous to (VII 5.10) holds. 55

VII 6. SOBOLEV FUNCTIONS AS THE INTEGRALS OF THEIR DERIVATIVES We begin with a statement for v = 1 to the effect that any function x of one real variable possessing first order generalized derivative y coincides almost everywhere with a function which is locally AC, and x' = y almost everywhere. This statement will be the converse of what we proved in (VII 2). (VII 6.i) If v = 1, if x(t), a < t < b, is an element of Wl(loc, (a, b)) 1 oc with generalized derivative y(t), a < t < b (thus, x,y E L (a, b)), then there is a function f(t), a < t < b, which is continuous in (a, b) and AC in every closed interval [a, b] c (a, b), such that x(t) = f(t), y(t) = f(t) a.e. in (a, b). In particular, if x c Wl(a, b) (hence, x, y C L(a, b)), then f(t), a < t < b, is continuous and AC in [a, b]. Proof. Let a, P be any two points a < a < f < b, and denote by n any integer sufficiently large so that n < a - a, n < b - 3. Let ~(t), t e E, be defined by taking 0 = 1 in [a, i], 0 = O otherwise. Let 0 (t), t E E1, be defined by taking 0 = 1 in [a, P], 0 = 0 outside [a - l/n, P + 1/n], 0 = n n n 1 + n(t - a) in [a - l/n, a], and 0 = 1 + n(P - t) in [I, P +1/n]. Then, 0, 0n are in L (E1), 0 is continuous, and both have compact support. Also, 0 is AC with bounded derivative 0' = n in (a - l/n, a), 0' = -n in (I, P + l/n), n n and 0' = 0 outside [a - l/n, a] and [I, P + l/n]. Also, 0 + 0 in L (E1) as n n n - oo, since fE lo - O[dt = 2(1/2n) = 1/n. 56

Note that, for any x E L (El) we have f x dt x dt as n +. (vII 6.1) E n E Indeed, E x( - n dt- I (+ fl/n )[xdt (VII 6.2) and the last expression certainly approaches zero as n -+ o Let us prove that b b fa Y n dt -= a x O' dt (VII 6.3) a n a n First, we know that fb y i dt = -b x' dt for every C C(a, b). Thus, a a d for any fixed n and E, O < E < 1, sufficiently small we have fb y(J n)dt = -Jb x(J En) dt. As e - O we know that J ~ + O n (Jn n)' -' a.e. in (a,b) a en e n nn n as n co, with 0 < (J n)(t) < 1, O < (Je n)t(t) < n for all t E (a, b). Thus, by dominated convergence theorem, we have f y(J )adt b fa y n dt, fb x(J )'dt * + b x V' dt as e - O, a en a n d a en a n and (VII 6.3) is proved. As stated by (VII 6.1) the first member of (VII 635) approaches f y b dt as n - 0o, or |f y dt. The second member of (VII 6.3) equals n f /n x dt - n -l/n x dt; hence, the second member of (VII 6.3) approaches x(B) - x(c) for all o, f 57

outside a possible set E of measure zero, or a, f e (a, b) - E. Thus, as n - oo, and a, 5 E (a, b) - E, we obtain from (VII 6.3) y y dt = x() - x(a). In other words, if t E (a, b) - E, we have x(t) = x(t) + f y(t) dt for all t c (a, b) - E. Thus, x coincides a.e. in (a, b) with the AC function f(t) = x(t) + f| y(t)dt, or x(t) = f(t), y(t) = f'(t) a.e. in (a, b). If y E L(a, b), then the last expression defines f as an AC function in [a, b]. A property (P) is said to hold for almost all intervals I = [a, G] c G, a = (a,...a),, = (,. ) a < Di, i = l,...,v, if P holds for all intervals [a, D] as above with (a, D) E E x E - E where E is a subset of E x E of measure zero. V To make this definition more precise, one may observe that the set of points (a, D) E E x E such that I = [a, ] c G, is an open set, namely, V V G* C G x G c E x E, and thus E is a subset of measure zero of G*. V V (VII 6.ii) If v > 2, if x(t), t c G, t = (t,...,t ), G C E is any function boc in L (G) with generalized first order partial derivatives y. = D.x, i = ( iL 1 l,...,v, (also in L (G)), then for almost all closed intervals [a, i] c G the following relation holds: 58

Proof. If is enough to prove this statement for v = 2, and then we can write (t, s) for (t, t ), [a, b; c, d] for (c, a ), xt, x for Yl, Y2. Then the proof is analogous to the one for (VII 6.i) where intervals I = [a, b, c, d] c G, I = [a - 1/n, b + 1/n, c - 1/n, d + 1/n] C G are used, and functions n 4, * defined by, = 1 for t I, I = O for t E - I; O = 1 for t E I, n v n 4n = 0 for t e E - I, 4 = 1 - n(t - b) for (t, s) E T = [(t, s)f b < t n v n n in < b + 1/n, c - (t - b) < s < d +(t - b)]J, etc. loc (VII 6.iii) Lemma. Let x(t), yl(t),...,y(t) t E G, be functions inL (G) such that relations (VII 6.4) hold for almost all closed intervals [cx,] c G. Let [A, B] be any closed interval [A, B] c G. Then for each i = l,...,v, there is a function fi(t), t e [A, B] such that (a) fi(t) = x(t) a.e. in [A, B]; (b) for almost all t' e [A:, Bi], fi(t., ti) is AC in ti on the linear interval [Ai, Bi]; (c) for almost all t' E [A:, B:] we have afi(t;, t )/at' = yi(t, ti ) where af.i/t is the usual partial derivative of fi(ti, t ) with respect to ti Proof. Let [A, B] and i be fixed. Then the relations!bb a [x(t:, b.) x(t, ai) dt f Yi(t) dt, i = l,...,v, (vII 6.5) for all intervals [a, b] c [A, B] such that the point (a, b) does not belong to a certain set E or 2v-measure zero in [A, B] x [A, B]. By Fubini's theorem, if the point (ai, bi), Ai < ai < bi < B, is not in a certain set F of 2-measure zero, then the set E(ai,bi) of points (ai,b!) such that (a,b) e E is of 59

- ti A' i t (2v - 2)-measure zero. Let H be the set of values t, A < t < B, for which x(t:, t ) is not integrable in t' on [A:, B:]. By Fubini's theorem H is of i' 1 i linear measure zero. Let I be the union of F and of the set of points (ai, bi), A. < a. < b. < B., such that either ai or bi is in H. Then, I is still of 2measure zero. Moreover, if (ai,bi) E [A,B] - I, then the integrals on both sides of relation (VII 6.5) are continuous in (a!, b') on [A:, B:], and hence relation (VII 6.3) holds for every interval [a, b] c [A, B] such that (ai, bi) is not in I. Now let a. be any number, A. < ai < B, such that the set of values b for which (ai, bi) c I is of linear measure zero, (such an a. exists by Fubini's theorem). Then ai is not in H, and we define fi by taking _ ti f.(t, ti) = x(t', a) + y(t t )dt ( 11 i' 1 1 (VII 6.6) a. i for each t Ec [A', B:] for which Yi(t:, ti) is integrable in t, on [Ai, Bi], (this being the case, by Fubini's theorem, for almost all t' in [A:, BI]. We define f. = 0 otherwise. Clearly, fi is measurable on [A, B] and AC in t on [Ai, Bi] for almost all t' [A', B']. Furthermore, for almost all t' E [A:, Bi], the first order derivative 3fi(t., ti)/ati exists and is equal to Yi(ti,ti) for almost all ti E [Ai,Bi]. Hence, we need only to show that f.(t) = x(t) a.e. in [A,B]. By integrating the right-hand side of relation (VII 6.6) with respect to t (this is possible by the choice of a), we see that f(ti, t ) is integrable 1 - P 4l - in ti for each t, and that the relation 60

B!' B'dt +B Bi 1 B i' ti IA1 Jfi(t, i' Jdtl < fA' Jx(t' aidti + Yi(t) Idt'holds independently of ti. Thus, by Fubini's theorem, fi is integrable on [A, B]. Moreover, relation (VII 6.5), with x replaced by fi, holds for every interval [a, b] c [A, B]. Hence, if b. is not in a certain set of linear measure zero, then f.(t, t ) and x(ti, t) are (by Fubini's theorem) integrable in t. Thus, by taking a = ai and b. as above in relation (VII 6.5), we have, by (VII 6.5), that bi' bid fX(t, bit = )a. 1 fi(t', bi) dtL ai i i i a. i i 1 for every [a, b'] c [A!, B']. Hence, for such a bi, we have x(t', bi) = fi(ti, bi) for almost all t' E [A', B!]. Since x and fi are both integrable on [A, B], it follows that fi(t) = x(t) a.e. on [A, B], and lemma (VII 6.iii) is thereby proved. 10c (VII 6. iv) Lemma. Let x(t), Yl(t),.,y(t), t E G, be functions in L (G) such that relations (VII 6.4) hold for almost all closed intervals [a, P] c G. Let [a, b], [A, B] be closed intervals with [a, b] C int [A, B] c [A, B] c G. For each i = l,...,v, let fi(t), t E [A, B], be the function defined in (VII 6.iii). Then (a) (J~x) (t) + x(t) = f.(t), (XJ x)/Vt )(t) + y,(t) as e - O a.e. in [a, b], i = l,...,v; (b) for almost all t' E [a!, b!] we have J x - i i' i S fi(t) as 0 * O uniformly for ti E [ai, bi]. 1 E i~~~~~~~i~ 61

Now (J x)(t) = A j (t - T)x(~)dt, for t c [a, b] and s > 0 sufficiently small. Henrce aJ x(t)/ti = BA(aj (t-T)/ti )x(T)dt = AB(aj~(t-T)/ti) fi(T)dt = _fB( j~(t —T)/ Ti)fi(T)dt Ai Ti f jA( jE t - i ii)/aTif (T i )di 1 i B B~(t-) Yi ()dt = (J~yi)(t), where we have integrated by parts in the interior integral by using (VII 5.iii). Now, by(VII 3.iii), we conclude that oJ x(t)/ti + Y i(t) as e + 0 a.e. in [a, b], i = 1,...,v. Part (a) of (VII 6.iv) is thereby proved. To prove part (b) let us assume first that Yi(t) is of constant sign on [A, B], say yi(t) > 0. Then fi(t!, t ) is continuous and monotone nondecreasing ti 1 with respect to t for each t! E [A', B.] not in a certain set Z1 of (v - 1)i i 1 dimensional zero. By (VII 3.iii) and (VII 6.iii, part (a)), J x(t) -* fi(t', ci) for all t' not in a certain set Z(ci) of (v - 1)-measure zero. Let S be a countable set of such values ci such that S is dense in some interval [ai, bi], A. < a. < b < B., with a., bi E S. Let Z2 be the union of Z1 and of the sets 1 1 1 1 2 1 Z(ci) with ci E S. Then Z2 is still of (v - l)-measure zero. By (VII 6.6) we see that for each t! E [A', B!] not in Z2, the functions J (t!, t ) as well as f.(t!, t ) are continuous and monotone nondecreasing in t. Moreover, for ti? Z we have J (ti, t ) - fi(t!i t ) for all ti E S, and hence, by the monotonicity and continuity, the convergence is uniform for t E [ai, bi]. 62

If Dx(t) changes sign on [A, B], then choose c, Ai < c. < Bi, so that. fi(ti, t ) is integrable with respect to t on [A, B], and define i ii i t + i i fi(ti t ) = fi(ti, ci) + f g.(t' T )dT, + 2. 2. 2. 2. c* gi =.,i C ti = i proof reduces to the case of the preceding paragraph. (VII6.v ) Theorem. If x(t), t E G is anelement of (loc,G), then there is a)d ACfor each t for which f(t ti) is C with respect to ti on the linear open set Gi(I); and (d) (tt)/at Dix(tt +) a.e. in G. In prticular, if w(G), Proof. Let x (t)g i )g = 2Jx)(t) as E O whenever this limit exists and finite, and set f. = O otherwise. Then f =(VII f - f on [A B], and as in the first inart of the proof, both f and f(VII are AC and monotome nondecreasing with i + - respect to t for almost all t. Since J x = J f = Jf i- fi' the proof reduces= 1,2,..., be twohe case of the preceding paragraph. (VII 6.v) Theorem. If x(t) t E G, isan element of Wl(loc,G), then there is a 1 function x (t), t ~G, such that (a) x EW (loc,G); (b) x (t) = x(t) a.e. in G; ( c) for every i = 1,...,v and for almost all t G. the function x (t') is AC with respect to t on the linear open set G(t ); and (d) x (t )/ti = o i0 D.x(tit) a.e. in G. In particular, if wEW (G), then x EW (G). 2. 2. 1 o 1 Proof. Let x (t) = lim (J x)(t) as E -+ 0 whenever this limit exists and finite, and set x = 0 otherwise. By (VII 3.iii) we have x (t) = x(t) a.e. in G, and parts (a) and (b) of (VII 6.v) are thereby proved. Let R, R'I m m m 1,2,..., be two sequences of closed intervals R = [a b ], = [A B], ~~~~~m m m m m m 65

t e R', be the functions defined in (VII 6.iv) in correspondence of the inm terval R' and the given function x. Then, for each m and i, by (VII 6.iii) m and (VII 6.iv), for almost all t' E [a'.,b'i], x (t, = f.(ti,t ) is AC a. ma. m. 0 ia. ma. iA in t on [ami,bmi], and x(t = a(t = Di(t',tl) for almost ma m fma a a. a.mos t all ti C [a i,bmi]. Also, f = f a.e. on R n R for any two a m m. *h mi mli m m1 R n R, and hence by continuity we have f i(ti,t') -f i(ti,t) for all t E [ami,bmi] n [a.,b. for almost all t' E [a'.,b'.] n [a'.,b i]. m —a.m ma. ma m1 l a m1 mi ml mli Since G is covered by the countably many intervals R, we conclude that, for almost all ti, x (t!,t ) is AC on Gi.(t!), and 2xi(ti,ti)/ti = D x(t',tl) for i almost all t E G.(t!). Remark. We are now in a position to state and prove the following statement which is the converse of (VII 6.ii): If x(t), yl(t),...i,y(t), t E G, are functions in L (G) such that relations (VII 6.4) hold for almost all closed intervals [ap,] c G, then x e W (loc,G) with generalized first 1 order derivatives D x = yi, i = 1,...,v, a.e. in G. Indeed, by repeating the argument in (VII 2.ii), we see that relation (VII 2.1) can now be proved with x replaced by f. and y replaced by afi/6t i Yi a.e. in G; hence relation (VII 2.1) holds for x and yi, i = l,...,v, that is, yi is the generalized first order partial derivative of x with respect to is, y. t in G, according to the definitions of (VII 2), i = 1,...,v. 64

VII 7. BOUNDARY VALUES OF SOBOLEV FUNCTIONS ON THE BOUNDARY OF INTERVALS We initiate here the study of boundary values of Sobolev functions. For the sake of simplicity it is convenient to begin with boundary values of such functions on the boundary of intervals. This will apply immediately to func+ 1 tions defined in E and their boundary values on the hyperplane t = O, by v 1l i i 1 considering arbitrary intervals [O < t< b, a < t < b, i = 2,...,v] c E V In the next section we shall introduce the concept of regions of class K, and then we will be able to define boundary values of Sobolev functions in such regions. Let R = [a,b] be a closed interval in E, and let R0 denote the interior V of R. Let x(t), t E R, be a function of class Wl(R ), p > 1, and let x (t), p 0 t C R, be the corresponding function defined in (VII 5.v). Then, for each i = 1,...,v, and almost all ti E [a',b'], the function x (t',ti) is AC in ti on the linear interval (a.,bi) and the limits exist:il(ti) = x (t,ai +o).(t = x (tjb-o). (VII 7.1) 11 i o 1 1 i2 1 0 i 1 Any change of values of x in a set of measure zero in R may imply a change of values of x also in a set of measure zero, but —as one could retrace from o the proof of (VII 6.v) and previous lemmas, and as we shall prove independently below —the limits (VII 7.1) may be altered at most in a set of (v-l)measure zero in [a,b!]. The 2v functions Oil' Pi2' i = 1,...,v, define —up to a set of measure zero on aR —a function ~(t), t e aR, on the boundary of R. We think of 0 as defining an equivalent class on bd R. We say that, is the set of 65

boundary values of x on aR. If O coincides a.e. on 6R with a function which is continuous on aR, we say that x has continuous boundary values. (VII 7.i) If x(t), t e R, is an element of W (R ), p > 1, then the boundary values ~(t), t C aR, of x are in L (RO). The function 4 will be often denoted by yx. Proof. It is enough to prove that O(t,ai) is L -integrable on the face 11 p F = (t. = ai, t E [ai,bi]) of R. To prove this, let us consider two numil 1 1 i i i bers til, ti2, ai < til < ti < b.. Then for almost all t! E [a',b'] we have o (tl't )- x (tl ) = f D x(ti,T )dT, (VII 7.2) o i i2 o i where x is the usual function defined in (VII 6.v). For p = 1 we have o b' Jaf dtIft D x(t!,T1)dT'j < K f f 2lD x(t)jdt. (VII 7.3) i 1i Since D.x is L-integrable in R, the last member certainly approaches zero as 1 til, t a.. This proves that the limit x (t',t ) d < (ta) as t it a occurs, not only pointwise almost everywhere on Ffl, but also strongly in L (F il). Thus, as til a,, we deduce from (VII 7.3) that fS ix (tetr2) (t!)Idt' < R f a12 ID x(t)cdt. (VII 7.4) i 1 1 For p > 1, we deduce from (VII 7.2) and Holder inequality that 66

ai Io ii2 o i il t i - b' t'dt'If D x(t',T)dT I| < It -t i tl a (t)I dt. ay ii 1 i i Thus, x (t',ti) il(t') as t i a. strongly in L (w. Aq t+ a, we deduce as before b' b' t J Ix (t't ) (t)lPdt < ti2t -ailP-1 a f i2 IDix(t)lPdt. 1 i i2 (VII 7.5) (VII 7.ii) If x(t), x (t), t E R, k = 1,2,..., are functions in Wl(RO), k p p > 1, if fRIDiXk(t)I dt < M for some constant M and all m = 1,2,..., if xk - x strongly in L (R), then k 4 ) strongly in L (aR), where *(t),'k(t), t E aR, denote the boundary values of x, xk. The same result is true if p = 1 provided the generalized derivatives Dixk(t), t E R, k = 1,2,..., 0 i = 1,...,v, are known to be equiabsolutely integrable on R Proof. Assume p > 1 and note that relation (VII 7.5) holds for x as well as for x. Then from (VII 7.5) and the uniform boundedness of the numbers Jab IDix(t)l b D x (t)IPdt, i = 1,...,v, k = 1,2,..., we see that, given E > 0, there is some 5 > 0 such that for each ti2 with ai < t < ai+6, we have i i2- 1 bf x1/p ai ix(tIti ) - (t')lPdt' < ~, i = 1,...,v, and fat i i2 ill i xk(ti'ti2) kil(t)lPdt < e, i = l,...,v, k = 1,2,.... 67

Since xk - x strongly in R, there is a k such that b,x(t)_-x(t)Pdt ]l/p < 51/P E for all k > k. By Minkovski's inequality we have now - O't 1/p bfi 1J (t') - O (t')lPdt', I a* a kfil (i kil(tldt idt a b' ~-1 a + b'. 1/p < o ~l/Ptra:, 1pil(t')-x(t{,t )l + IX(tit ) kti)I + Ixk(t.,ti) - kil(ti) dtidt + f ai /P (t )-x(t' ti) Pdt'dti} a, a k l 1 i < 56i f +f Ix(t' t )-xk(t' )Il ai' i'i k i + Ix'(t<yfai fJ ( (tjtdtt1/ ilkilPkil i { a 1 (t'-.i Jit ) Pldtidtdt f~~kil i na + 5 in L(Fil). If p =1 and the functions Dxk, i =l,...,v, k =1,2,..., are equias well as for Xk, we see that given ~ > O there is a o > v such that for 68

each ti2, ai < ti2 < ai+,we have Ifa i2 i2 i failX(t',ti) - i(t)ldt' < e. a' i i2 kil i i i Also, for some k and all k > k we have O -O b f Ix (t)-x(t)Jdt < 6E. a k The details are now analogous to those of the previous case, and the statement (VII 7.ii) is thereby proved. The case p = 1 in statement (VII 7.ii) is actually exceptional, as it can be seen by the following example. Let v = 1, p = 1, xk(t), 0 < t < 1, be defined by taking xk(t) = l-kt for O < t < k, xk(t) = 0 for k < t < 1. Then lixk|l < 1, IIx'll = 1, k = 1,2,..., and xk E W (I) where I = (0,1). If x(t) = O, 0 < t < 1, then xk + x strongly in L1(I). On the other hand k(~) = 1, ~(o) = 0, and k does not converge to ~(o). Statements (VII 7.i) and (VII 7-ii) extend immediately to functions z E Wm(RO), p > 1. Indeed we have p (VII 7.iii) If z(t), t E R, is an element of Wm(Ro), p > 1, m > 1, then z p and each of the generalized partial derivatives D z with 0 < Jua < m-l, possesses boundary values 4 (t), t E R, and O E L (R). p (VII 7.iv) If z(t), zk(t), t c R~, k = 1,2,..., are functions in Wm(R ), p > 1, m> 1, if ROIDazk(t)IPdt < M for some constant M and all 0 < JaJ < m, 69

and k = 1,2,..., if D Zk - D z as k + o strongly in L (R ) for all 0 < Ial < m-l, then k U strongly in L (SR) where O (t), Ok(t), t E 6R, denote k p k the boundary values of z, zk. The same result is true if p = 1, provided the generalized partial derivatives D zk(t), t E R, with ai = m and all k = 1,2,..., are known to be equiabsolutely integrable in R~. We are now in a position to prove a statement similar to (VII 4.iv) which we shall use in (VII 10): (vII 7.v) If xE W (E ) and all partial derivatives D X, 0 < lai < m, have P v boundary values 7D x = 0 on t = O, then there is a sequence of functions xk - W (E)) n C (E ) with lIxk-xIm as k kP v ov k p Proof. As in the proof of (VII 4.iv) we take h = (h,0,...,o) with 1 1 h > 0, and we denote by y the function defined by y(t) = 0 for t' < h, y(t) = x(t-h) for t' > h. Let us prove that y E Wm(E.) with Da (t) = 0, for p v 1 a a 1 1 t < t, D y(t) = DUx(t-h) for t > h It is enough to prove that corresponding relations (VII 6.5) hold for all Da 0 < i aI< m-l, and their first order partial derivatives. This is trivial, of course, up to a displacement, a 1 1 and the use of the boundary values 7D Y which are all zero on t =h. Now we know that |ID y -D X IE - 0 as h + 0+, 0 < jaj < m, and, on the other hand, for any h > 0 fixed, we also have |D a(J y) -DyY' p 0 as ~E 0+, 0 < jaj < m. _ p For suitable values of h hk E =k >, we can now take xk =J y. 70

VII 8. INVARIANCE OF SOBOLEV FUNCTIONS WITH RESPECT TO TRANSFORMATIONS OF CLASS K 1 v (VII 8.i) Theorem. If z(x), x = (x,...,x ) E G, is a given function z e W (G), p > 1, and T: U + G, or T: x = x(u), u = (u,...,u ) E U, is a transp formation of class K (see (VII 2)), that is, T is one to one and Lipschiztian -1 together with its inverse T: u = u(x), x E G, then Z(u),= z(x(u)), u E U, is a function Z e W (U), and Dui Z(u) = ZV= D j x(x(u)) D i x (u), i = 1,...,v, a.e. in U. j=l x U Proof. First assume p = 1. We have JTu- Tvf < Kju - vI for all u, v E U and some constant M; hence the functions xi(u), j = l,...,v, are Lipschitzian in u with the same constant K, and finally ID i x(u) _< K a.e. in U, i,j = l,.~.,v U (see (VII 1.i)). Then we have also Idx/dul < M a.e. in G,where dx/du is the Jacobian, and the constant M is certainly < v'K v Let us consider a sequence Rk, k = 1,2,..., of open sets invading G, that is, Rk C Rk+l, RktG, the closure of each Rk being the finite union of closed intervals in G. For each k we shall consider also an analogous open set R' such that Rk ccl R C RI C k k k k cl R C G, the closure of each Rk being again the finite union of closed intervals in G. By (VII 4.iv) we know that there is also a sequence zk(x), x E Rk, k = 1,2,..., of functions, each zk continuous in Rk with its first order partial derivatives D i zk(x), x E Rk, i = l,...,k, such that Rk[ k Z IDi z - D zkld < /k, k= 1,2,..., 71

and actually each zk is continuous with its partial derivatives in the open neighborhood R' of R (indeed, we can take z = J z for e < dist(R', aG)) k k k k and ~ sufficiently small. If zk(u) = z'(x(u)), u C = T-1(R'), then Zk(u) is a continuous function I k u) =z x(u)), u k kSk of u, and for every i and every u' the function Zk(u, ui) of ui only is AC with respect to u in the one-dimensional open set Sk(u~) uo all u such that (u.', u) E Sk. Indeed, the same functions are AC as superpositions of a function which is continuous with all its first order partial derivatives, and a Lipschitzian function, and k u', Sti i function, and 2ak(ui' u )/au exists for almost all u e Sk(uI) and admits of a bound which is independent of u. and u' (but may depend on k). Thus, Z E 1 k W (S'), and if S = T (Rk), we have S C clSk Sk c cl S'c U. Since clSk 1k k k k k k k is a compact subset of S', by (VII 4.iv) we know that Zk(u) and its first order partial derivatives are the uniform limit on Sk of the mollified functions J Zk(u) and their first order partial derivatives. For every kwe shall take, therefore, an E = ~(k) > 0 sufficiently small so that, if V J Z (u), ~ = ~(k), u c Sk, we have S [Iv - Zl I+ IDi Vk - D.Z J]du < l/k, k 1,2,..., (VII 8.1) where, by force of (VII 4.i), we have iD Z k D zi (x(u)) = E D jzk(x(u)) D xi(u) k I k ij jk i u u x u j D jz(x(u)) ID xj(u) +Z[D jz (x(u))- D *z(x(u))]D xIj(u). 72

Hence, for k < p < q, D Z (u) - D iZ (u) = [D z(x(u)) - z (x(u S ID i p(U) - D iZ (u) Idu < K IS D z p(x(u)) - D jz (x(u)) du = K Rf IDx z (X)) D jz (x) Jdu/dxldx k x q < K MZ. f I[D.z -D.z I + ID z - D.zI]dx; k xi P x xj < K M(1/P + l/q) < 2K M/p. (VII 8.2) Analogously we have, for k < p < q, f Iv (u) - V (u) I < [Iv (u) - z (u)I + Iv (u) -z (u)! Sk P q Sk P p q q + Iz (u) - Z (u) ]du p q 1< /p + l/q + JRlz p(X) - Zq(X)I du/dxldx k < 1/p + l/q + M fR IJzp(x) - z (x) dx Bkp q < 1/p + l/q + M fR [Jz - zJ + z - z l]dx k < l/p + l/q + M(l/p + l/q). (VII 8.3) By combining (VII 8.1), (VII 8.2), (VII 8.3) we see that for every k <p <q we have U U U U + i D zP - D iZ J]du 73

< (M + l)(l/p + l/q) + 1/p + l/q + K M(1/p + l/q) < 2(M + 2 + KM)/p = Mo/P Hence, there are functions V(u), Qi(u), u E U, i = l,...,v, such that fSkI - VI i DV - Qij]du < Mo/P for all p > k. (VII 8.4) Sk By combining (VII 8.4) with p = k and (VII 8.1) we have S [IZk- Vi + ID iZk- QiJ]du < (M + 1)/k, k u for all k = 1,2,,,, where D Zk(U) = D Zk(X(U)) = D z k(X(u))D xj(u) a.e. in Sk, i ( ik jk i k U U x U and i = l,...,v. Relation (VII 8.4) for p = k and k = 1,2,..., implies by force of (VII 4.iv) that V is an element of W (U) and that Qi' i = l,...,v, are the generalized first order partial derivatives of V, or Qi = D iV, i = l,...,v. U On the other hand, for every interval S C U and m sufficiently large so that S c Skwe have now f IQ (u) - E. D jz(x(u)) D iJ(u) Idu x u -< fSIQi G- D iZkIdu + fSID iZk E.D jzk(X(U)) D ix3(u) Idu S (X(U)) DZ(X(U)) D x (u)u + fs |ID xz(x(u)) - D iz(x(u))l D xu(u)du x x U (M + 1)/k + 0 + K Z f ID z (x(u)) +D z(x(u))Idu x x < (M +l)/k + K/k = (KM + M + 1)/k, 74

where m can be as large as we want. Thus, the integral in the first member is zero and Dz(x(u)) Q (u) Z D kz(x(u)) D xJ(u) U x U a.e. in S and hence a.e. in U. Statement (VII8.i) for p = 1 is thereby proved. The proof for p > 1 is analogous, and is left as an exercise for the reader. Theorem (VII 8.i) can be expressed by saying that functions of classes W (G) are invariant with respect to transformations of class K. In particular, p they are invariant with respect to transformation of class C1, or of change of orthogonal coordinates, and finally, by force of (VII 4.iv), with respect to passage from Cartesian to polar coordinates. We shall now state an extension of (VII8.i) to functions z E Wm(G. We shall consider one to one transformations T: U - G, or T: x = x(u), u = (u,...,u ) e UI with inverse T: u = u(x), x = (x,...,x ) E G, such that all functions D xJ(u), u E U, and D uJ(x), x E G, j = l,...,v, 0 < lal < m-l, exist and are continuous in U and G respectively, and the same functions with Jao = m-l are uniformly Lipschitzian in U and G, respectively. We shall say that T is a transformation of class K. Thus, the transformation of class K1 m are the usual transformations of class K considered above. The following theorem holds: (VII 8.ii) Tfz(x), x = (x,...,x ) e G, is a given function z E Wm(G), m > 1, p> 1, and T: U - G, or T: x = x(u), u = (u,...,uV) C U, is a transformation of class K, then Z(u) = z(x(u)), u 6 U, is a function Z e Wm(U) and usual ~m ~~~~7p 75

formulas for the partial derivatives D Z, 0 < lojS < m, hold almost everywhere in U.

VII 9. OPEN SETS OF CLASS K We shall now introduce the concept of open set of class K, or K1, and of class K, m > 1, in the t-space E, t = (t,...,t ). We shall often denote m -- v these sets as regions of class K or K m A bounded open set G c E is said to be an open set, or region, of class K or K1, if (a) G can be covered by finitely many open sets y, s = 1,...,N; (b) If I denotes the interval I = [O < u < 1, -1 < u < 1, i = 2,...,v], there is an N', O < N' < N, and for each s = 1,...,N' a positive transformation T of class K (see (VII2)) defined on I such that y = T (I), s = 1,...,N'; (c) If X denotes the segment A = [u = 0, -1 < u < 1, i = 2,...,v] and I' denotes the interval I' = [O < u < 1, -1 < u < 1, i = 2,...,v] = I U X, then for each s = N'+1,...,N, there is a positive transformation T of class K defined on I' such that y = T (I), r = T (x), and r C aG; (d) the sets r., s = N'+1,...,N, form a finite cover of aG. Note that the sets (r = T (I), s = 1,...,N', 7' = T (I') = y U Frs s = N'+l,...,N) form a finite cover of cl G. Note that each part r of the boundary of G is in one-to-one correspondence with the(v-l)-dimensional cell X and this correspondence is certainly Lipschitzian with its inverse. If G is of class K, then we can say that cl G is a v-dimensional manifold with boundary of class K, and that ~G is a (v-l)-dimensional compact manifold of class K. In both cases "of class K" expresses the fact that only transformations of class K are used (one-to-one and uniformly Lipschitzian with their inverse. Any system [7, T, s = 1,...,N] as above will be said to be a typical representation of G as a "region of class K." 77

If in all definitions above we use only transformations of class K, m > i, (see (VII 2)), then we say that G is a region of class K in E m v Let G be a region G C E of class Km, m > 1, and let [ys,T,s = 1,...,N] be a typical representation of G. Thus, [y1'...'N] is a covering of G, and [Ki' i = 1,...,N', U F, i = N'+1,...,N] a covering of clG. We can always think of 7i U ri i = N'+1,...,N, as a part of an open set y with i - P. i E -G, i = N'+l,...,N, so that [1,..'N,, YN,+'1... N] is a covering of clG. We shall now consider a partition of unity [1''N]' where y C C (E ), y has compact support K c y for s = 1,...,N', y has compact sup0 V s S S S port K c' = y + r for s = N'+l,...,N, and ZN 1 (t) = 1 for every t EclG. If x(t), t C G, is of class Wpm(G), then we have x(t) = Zsl x(t) 4s(t), t c G, (VII 9.1) s=l s and the same relation holds even on aG whenever x(t) has boundary values on?G. Note that x(t) 4 (t), t c G, has compact support K c K c y if s = 1,..,N', and compact support K C cl G n y' = y U r if s = N'+l,...,N. By force of (VII 4.ii) the functions x(t) 4s(t) are of class Wm(G) and their p derivatives are given by the usual rule Da(x s) = a () DxD a-s s 5 where a (al.9'''av), 0 < I a < m, B = ($1,..,v) and Z ranges over all f with 0 < B << a, i = l,...,v. i 1 Thus, if we choose a given representation of G and a corresponding partition of unity, then we can find constants K, K' such that 78

ID (x *s)(t)l < K IDJ1 < IjIDPx(t) a.e. in G, IID (x S) )ilp K' 1 < jj IDC for all s = 1,...,N. Finally, T I y, s = 1,...,N' T: I' -* U r, s = N'+1,..,,N S s S and we consider the functions Z (u) = T- (x(t) s(t)) (VII 9.2) s =1,...,N. For 1 < s < N' these functions Z are of class Wm(I), and have com_ -S p pact support K I. For N'+l < s < N these functions Z are of class W (I) but have compact support K cI'= I UX. In either casewe can extend these functions + + + M + in all of E by taking them equal to zero in E -I, or E -I', andZ EWm(E). V V V S p V I +I If the functions Z (u) C Wp(E ) have continuous boundary values 7Z on s ~pV S the cell X, then the corresponding functions x(t) 7 (t) have continuous boundary values y(x(t) *s (t)) on r, and so x(t) has continuous boundary values yx on aG. The same holds for the derivatives D x, 0 < WJ < m-l. If these boundary values are not continuous functions, we must show that they are measurable with respect to the natural hyperarea measure a on the boundary ~G of G. This hyperarea a can be introduced rather straightforwardly 79

by means of the following statement. (VII 9.i) If G is a class of K in E, m > 1, then there are completely additive set functions Si(E), Vi(E), i = 1,...,v, and a(E), defined on a suitable class 6 of sets E c aG, with the following properties: (a) Vi(E) > O, a(E) > O, ISi(E)J < Vi(E) < a(E), i = l,...,v, for all E E 6. (b) If a(E) = 0 and Ei denotes the projection of E on the hyperplane X. Iti = O] of E, then 4EiE = 0O where 0Ei denotes the (v-l)-dimensional Lebesgue measure on Xi, i = 1,...,v. (c) If [y, T, s = 1,...,N] is any representation of G, and T: I U X Y u r, s = N'+1,...,N, hence T maps X onto r c ~G, and F is any measurable subset of A., then E = T (F) e ~, and Si(E) = IF(dt'/du')duI, Vi(E) = IF Idt/du 1dU, i =, a(E) = F[[I=1(dt/du) 2 1/2 du{, (vII 9.3) where tV = (t,,ti-l, ti+l tv) u = (u2,...,u ) and dt'/dul, i I P V i = 1,...,v, are the usual Jacobians of the transformation T. S S (d) If T: cl G + cl H is a transformation of class K, then sets measurable m with respect to a on aG and to a' on aH correspond, and there exists a constant K > 1, depending only on the transformation, such that 8o

-1 K a(E) < a'(E') < K o(E) whenever E on aG and E' on 2H are corresponding measurable sets. Proof. If E, F, T are as in (c) above, let T. denote the natural projection operation of E onto the hyperplane Xi = [tlti = O] of E. Then T.T: A + Xi, or tV = t'(u), u E X, certainly is a continuous mapping nonI S i s necessity one-one, from the (v-l)-dimensional cell X = [u = 0, -1 < u < 1, j = 2,...,v] into the (v-l)-dimensional hyperplane Xi = [-X < t < a, h i i, h = l,...,v, ti=0], and TiT is certainly Lipschitzian on A of the same constant M as T. By force of (VII 2.iii) with v-l replacing v, the set E' = S -1 vi(E) = TiT (F) is measurable. We take ai(E) as given by first relation (VII 9.3), as defining a signed measure function. We must show, however, that such a definition does not depend upon the representation. Indeed, if [th' Th' h = 1,...,N] is any other representation of G, and we assume that for a given subset E of aG, we have E c c G, E c rh c aG, Ei = E = T T(F), Bi iT i h T.T Vt = t!(u), u E A, iTh t =.(U, u E Is i i i If t is any point of E, then t is an interior point both of r1 and rh, and we -1 - — 1 - denote by u = T t, u Th t the counterimages on X and X, respectively. We N3 t can even take a neighborhood N of t in cl G, such that N c s U, Nc h U h', and we denote by U and U the corresponding 81

counterimages U = T N, U = Th N. Now Th T is a one-one transformation of S h hs U onto U, certainly of class K (and T- Th is a one-one tranformation of class K of U onto U). Then Th T and T Th are positive transformations (as h s s products of positive transformations), and they certainly induce positive transformations between A and K, or du'/du' > 0 a.e. in K, and du/dul > 0 1 1 1 a.e. on A, and the same must hold, in view of (VII 4.iii), a.e. in F and F respectively. By (VII 5.i) we have then dt: dt du' dt' du' 1t= d * du = = d *'dUl = iF d 1- dudU S.(E) I du du - -- du i F du. 1 F du du 1 Fdu' duL 1 1 - dt' F du' S F d 1 s)i' This shows that S.(E) does not depend upon the representation (and even for the same representation we may choose any of the overlapping neighborhood elements r ). It is clear now that all these set functions join up to form a completely additive set function S.(E) over the whole of dG. The same argument, with obvious simplifications, holds for V.(E), i= 1l,...,v, and for a(E). The formulas in (a) for any representation follow by addition. Part (a) of (VII 9.i) is thereby proved. If E is as above and c(E) = O, then the v Jacobians dtt/dul, i = 1,...,v, must be zero a.e. on F, and this in view of (VII 4.iii) again, implies I FI = O, and finally IEI = 0, in view of (VII 2.iii). This proves part (b) of (VII 9.i). Part (c) was proved above. Part (d) is, a consequence of (VII 4.ix) on the multiplication rule for Jacobians. Statement (VII 9.i) is thereby proved. 82

We are now in a position to prove the identity Hm(G) = Wm(G) for any P p region G of class K, and to prove for such regions the interpretation of Hm(G) we have mentioned after relation (VII 4.4). All this is a consequence of the remarks already made at the end of (VII 4), and of the following statement. (VII 9.ii) If x c W (G) with G of class K, 1 < p < +oo, then there is a p m sequence of functions xk E W (G), k = 1,2,..., and each xk is Lipschitzian k p k in G together with all generalized partial derivatives D xk, 0 < jaj <m-l hence the derivatives D xk with lal = m are bounded, and IIxk - xl -P 0 as ~k~~~~k k pa If in addition, the boundary values 7D x are known to be zero a.e. on the boundary 2G of G for 0 < _aJ < m-l, then we can choose the sequence xk so as each xk is identically zero on and near aG. Proof. If we consider as at the beginning of this section any given representation [7s, T, s = 1,...,N] of G and a corresponding partition of unity s', s = 1,...,N, then we know that the corresponding functions Z (u) m + + belong to W (E+). We can now apply (VII 4.v) to each function Z (u), u c E, pV s to obtain a sequence Z k(U), u C E k = 1,2,..., of functions of class C (E ) such that IZs - 11m -* o as k * C. Finally, the functions Xsk = T (Zk ) have ks sps sp s ks the required properties on each y and r and we define them to be zero everywhere else on cl G. Finally, the functions xk = have the required properties onG U'G. 83

If the boundary values y D x, 0 < 7I1 < m-l, are all zero, then the + 1 functions Z (u), u c E, have boundary values zero on the straight line u = 0 s VI and so have the derivatives D Z (u), 0 < l1a < m-l. We can apply (VII 7.vi), and thus we can choose each sequence Zk, k = 1,2,..., so that all DCZsk, < la7 < m-1, are identically zero on u = O. Then the functions Z k and their derivatives DZk, < jal < m-1, are zero on and near 2G. 84

VII 10. WEAK COMPACTNESS IN L FOR p > 1 As seen in Chapter 4, we need compactness theorems in L (G) and Wm(G) p p for p > 1 as well as for p = 1. For the convenience of the reader we state and prove below some of these theorems for L (G) with p > 1 and with p = 1, and in(VII 11) we shall state and prove corresponding compactness theorems for Wm(G) with p > 1 and with p = 1. p We shall denote by t the real vector variable t = (t,.1,t ) E E, and V by z(t), t e G, a real-valued function defined on a subset G of E. By the V notation z c L (G), p > 1, we shall mean, as usual, that z is measurable and that IzlP is L1-integrable in G. (VII lO.i) Let G be a measurable bounded subset of E, and zk(t), t e G, k = 1,2,..., a sequence of real-valued measurable functions such that (a) Zk e L (G), k = 1,2,..., () fG IzkIjpdt < M, k = 1,2,..., for some constants p > 1, M > O. Then there is a measurable function z(t), t E G, and a subsequence [Zks] such that z e L (G); (VII 10.1) p lim f azldt < s- / Izk JPdt; (VII 102) G G s lim z p dt = lim Zk dt; (VII 10.3) G G s for every real-valued measurable function cp(t), t E G, with cp e L (G), l/p + l/q = 1. All integrals above are finite. 85

Remark 1. This theorem is usually proved as a consequence of general statements of functional analysis. Indeed the space L (G) with the usual L -norm is a uniformly convex normed space and hence symmetric by remarks of J. A. Clarkson, and consequently any strongly bounded sequence is weakly compact by a theorem of L. Alaog concerning weak topologies in normed linear spaces (see E. Rothe, Pacific Math. Journ. 3, 1953, 493-499). Nevertheless, there are direct proofs of statement (VII l0.i) and of the corresponding statement for p = 1 (see (VII lO.ii) below and its proof), and these direct proofs are based on the remark that hypothesis (P) implies that the function xk are equiabsolutely integrable in G, that is, given e > O, there is some 5 = 6(E) > 0 such that H C G, H measurable, IHI < 6, implies IH Izkldt < E, k = 1,2,.... Indeed, by Holder's inequality / Izkldt < (, lqdt) l/q( IzklPdt) 1/p M1/p IHI 1/q H H H and it is enough to assume 5 = sq M q/P Remark 2. There is a statement underlying (VII 10.i), namely that, for G bounded, (or at least G with finite measure IGI < + oo), the integrability of IzKP, p > 1, implies the integrability of every power Izr, 1 < r < p. This statement is not valid for G unbounded, as simple examples show (see for instance [24r]). All statements of this Section VII 10 have a slightly modified counterpart for the case in which G is unbounded. For the sake of simplicity we limit ourselves to the case G bounded, and we refer the interested reader to [24r] for extensions. 86

Remark 3. For p = 1 statement (VII lO.i) is not true, as the following well known example shows. Take v = 1, G = [o,l], xk(t) = k for 0 < t < k, xk(t) = 0 for k-1 < t < 1, k = 1,2,.... Then we can take x(t) = 0 for all 0 < t < 1, and now for c = 1, 0 < t < 1, we have I z dt = 1, t z dt= 0, and _.._ o k o (VII 10.3) is not valid. For p = 1 and G bounded, statement (VII lO.i) can be replaced by the following statement (VII 10.ii). (VII 10.ii) Let G be any measurable bounded subset of E, and xk(t), t e G. k = 1,2,..., a sequence of real-valued measurable functions such that (7) the functions xk are equiabsolutely integrable in G. Then there is a measurable function x(t), t E G, and a subsequence [xks] such that x E L1(G), (VII 10.4) 1im Jxdt < S IXkldt (VII 10.5) -- s IX t.ks G G fxcpdt = limf xkspdt, (VII 10.6) G s-o G for every measurable bounded functions cp(t), t C G. All integrals above are finite. Remark 4. Since G is bounded and has, therefore, finite measure, condition (y) certainly implies fG Ixk dt < M', k = 1,,..., for some constant M', andthus a condition analogous to (p) of (VII 10.i) is superfluous here. Proofs of (VII l0.ii) and of (VII l0.i). It is note restrictive to assume that G is contained in the hypercube 0 < t < N, i = 1,...,v, for some integer 87

N. Let us define each function xk in E by taking xk(t) = 0 for t e E -G. V k v Then the function xk are L -integrable in every interval R c E, and R Ixk (t)jdt < M, k = 1,2,.... Given any interval R = [a,b], a = (a,...,a), R k a_... b = (b,..,bV), by fa xk(t)dt we shall denote the integral of xk in R with the usual conventions concerning signs. Let R be the interval [o,N], 0 = (o,...,o), N = (N,...,N), N > O. For every k = 1,2,..., let us consider the function Xk(t) = f Xk(T)dT, defined for every t = (t,...,t ) E R, and k 0ko 0 where the integral ranges over the interval [o,t]. Then, for every interval R c R the interval functions o vk(R) = Xk( )dT, k = 1,2,.., R = [a,b] C R, k k o R can be expressed in terms of the usual differences of order v of the function Xk with respect to the 2v vertices of R, say Tk(R) RZk = Zk(b) - Zk(a) for v = 1,'k(R) = ARZk = Zk(bl,b2) - Zk(a,b2) - Zk(bl,a2) + Zk(al,a2) for v = 2, etc. As a consequence of (y) the interval functions k (R) = ARZk, k = 1,2,..., are equiabsolutely continuous in the usual sense, that is, given ~ > O, there is some 6 = 56() > 0 such that, for every finite system R1,...,RJ of nonoverlapping intervals Rj C Ro, j = 1,...,J, with EjlRjl < 6, we have If t, t' R, t = (t,...,t ), t = (t',...,t' ), and t-t' = d, let 88

pi, i = O,l,...,v, be the v+l points p = t, t, = (t, t 0 v 1 t' v-i+l,...,t' ), i = 1,...,v-l. Note that Xk(t ) - xk(t) = i=1 [Xk(Pi) k(Pi-l) i=l i k i: ~ - fiKL] xk(T)dT, and that the two intervals [o,Pi ] [~Pi1 ] differ by the single interval ri = (qi,Pi), where v-i+l qi (o,...,o,t,o,...,o), Pi = (t,..,t,t',...,t' ), and hence Xk k) - xk i=l Jq Xk i where rI <N - tv-i+l- t-il < NV It-tl= N d. Thus, given E > 0, v-l dv<l v-i whenever It-t'I = d < s/N, we have IXk(t)- Xk(t')l < Nv- (E/N ) = E for k k every k = 1,2,.... This shows that the functions Xk(t), t c R, k = 1,2,..., are equicontinuous in R. Since Xk(o) = 0, the same functions are equibounded in R. By Ascoli's theorem there is, therefore, a subsequence Xk, s = 1,2,..., with ks - oo, which is uniformly convergent in R toward a continuous function X(t), t c R. Since Xk(t) = 0 for every t = (t,...,tv) with t E R and t t...t = O, we deduce that X(t) = 0 for the same t. 0 For any interval R = [a,b] c R, we have 89

f Zks(T)dT ARXks = + Xk (aX), R where E. ranges over the 2 vertices a. of R with the usual sign conventions Ji J as mentioned above. As s -+ oo we deduce P(R) = lim zks (T)dT = X jX(a s-oR R and the convergence is uniform with respect to R c R. Since the interval functions Tk(R) = ARZk are equiabsolutely continuous in R, then the interval function T(R) = ARX has the same property. By Banach's theorem (V 6.ii) there is a measurable and Ll-integrable function x(t), t E R with T(R) = ARX = f x(T)dT R for every R c R, and lim f Xk (T)dT = f x(T)dT. s-oo R R This relation proves (VII 10.6) for every function cp which is the characteristic function of an interval. Thus (VII 10.6) is proved also for functions cp which are characteristic functions of a finite union of nonoverlapping intervals. If E is any measurable set, we can approach E in measure by means of a sequence of finite unions of nonoverlapping intervals, and then (VII 10.6) can be proved for functions cp which are the characteristic functions of measurable sets. Then (VII 10.6) is proved also for measurable step functions p. Finally, any 90

measurable bounded function cp can be approached by means of a sequence of measurable step functions with the same bound and thus (VII 10.6) can be proved in general. Relation (VII 11.5) is now a consequence of (VII 10.6). Indeed, if cp(t), t e G, is defined by taking p = 1 if x > O, c = -1 if x < 0, then p is bounded and measurable, hence by (VII 10.6) IxIdt = x pdt = lim J xks cpdt < s I XkS Idt. G G s-o G G Thus, statement (VII l0.ii) is completely proved. Let us now prove (VII lO.i). First, let us prove that for every c E L (G) the product xcp is Ll-integrable. Let TN (t), t E G, be defined by taking =N = if -N < N, WN = -N if cp < -N, where N is any integer. Then cpN is measurable and bounded in G. Also, let *(t), t E G, be defined by taking 4 = +1 if xcp > 0, f = -1 if xcp < O. Thus, 4 also is measurable and bounded in G, and so is cpN*. Note that pN and cp have the same sign, that lxi IT NI = IXNp I = XCNT, and hence, by force of (VII 10.6) fJ IXI (p dt = Xp N dt = lim j xks PNA dt, G G s-oo G where If Xks TN* dtj < f Ixksl pldt <( (f Xks)P dt) /P ( lqdt)l/q G G G G < M1/p (jf Iqdt)1/p G Thus J IJz I| N dt is below a fixed number independent of N. By Lebesgue G monotone convergence theorem, we conclude that zcp is L-integrable and that 91

f |Jxldt < M (f Ilqdt)lfq G G + _ If E, E denote the subsets of G where zcp > 0, or zcp < 0, respectively, then + E and E are measurable. By Lebesgue monotone convergence theorem we deduce J cp dt = lim J xcNdt, E N-oo E and by addition also f x p dt = lim f xcpdt. G N-oo G Thus, given E > 0 there is some N such that If x cp dt - J xNp dtj < E for N > N. (VII 10.7) G G Also If Xk pdt - Xks N dtl = If xXks(p- pN)dtl G G G < (f IXk Pdt!/P (/ i-Nqdt) /q < M( dt)/q G G G where the last expression certainly approaches zero as N - ~. Thus, we can take N such that o If Xks pdt - I Xks PN dtl < E for all N > N and all s. G G-0 (VII 10.8) Finally, from (VII 10.6) we have f XpN dt = lim xkcps N dt G s-xo G 92

for every N, in particular for N = N, and hence there is some s such that 0 0 If XCPN dt - f xks TN dtl < E, (VII 10.9) We have now, from (VII 10.7), (VII 10.8), (VII 10.9), Ij x pdt - f xks pdtI < 3E for all s > s. G G We have proved that relation (VII 10.3) holds for every c e L q(G) with 1/p + l/q = 1. Let us prove that x cP dtl < M1/P(/I Iqdt)1/q G for every p E L (G). Indeed, for any such p, If xcdtl = Ilim ksdt < pdtj < (I k Pdt) dt/ G s-*oo G G G < M/ (fI Plqdt)1/q G This shows that fG xTdt is a continuous linear operator in L (G) (and thus x C L (G)). This can be seen as follows. For every N > 0 let TN(t), t E G, be defined by taking N = Jx(t)JP-1 sgm x(t) if Ix(t)l < N; TN(t) = NP-1 sgm x(t) if Ix(t)I > N. These functions TN are bounded and measurable, hence I/ xcpN dtl:< Ml/P (f INl dt) l/q G G On the other hand 95

If XqNdt = xcPN dt t= r |xi PN jdt > f IcpI 1(PNIP-1 dt G G G G = I NIpp- dt = f JPI dt G G Therefore J IN qdt I </p (f ICNjqdt)/q ( NIqdt)l/Pdt M1/P, G G G for every N. Since QcN I - Ixlp-l, by Lebesgue monotone convergence theorem we conclude that 1xl(P-1)q is L-integrable in G, thatis, jxlP is L-integrable, and Jf xlPdt < M. G Now we can prove (VII 10.2). Indeed, if we take cp = xlP-1 sgm x, then cp is measurable, and I1 q = 1xlq(p-l) = Ix|p is L-integrable, that is, cp e Lq(G), and by (VII 10.6) f izlPdt = zcdt = lim f Zk cpdt G G s-oo G -lim k dt) / (/ IxPdt)l/qk G G and finally, by algebraic manipulation, Jf |zPdt < lim Jf dt. G s-vo G ks 94

VII 11. WEAK COMPACTNESS IN Wp(G) FOR p > 1 AND G OF CLASS Ki, AND SUMMARY OF THE EMBEDDING THEOREMS FOR Wp(G) We have seen in (VII 9) that every function x e Wm(G), 1 < p < + 0o, where G is a region of class K (see VII 9), can be decomposed into the finite sum x = ~ z = Z x4 of functions z whose transformations Z = ss 5s s s s T-1 (x* ) can be interpreted as elements Z of Wm(E+). It is now clear that all properties of the spaces W (E ), 1 < p < + 0, of (VII 5), (VII 6), p v (VII 7), can be transferred to the spaces Wm(G). We simply summarize below P the main results. (VII 1l.i) If x(t), t e G, is an element of Wm(G) and G is a region of class K in E, v > 1, p > 1, m > 1, then the generalized partial derivatives D x, m v - - O < Jal < m, are certainly of class L (G), and the boundary values yD x for 0 < Ica < m-l, are defined a-a.e. on aG and are certainly of class L (aG). However: (a) Each derivative Dax, 0 < Icx < m-l, is actually at least of class L (G) for every q, 1 < q < + 00, with l/q > l/p - (m-IlaI)/v, and ilDGll <KZD 11D' I|q: _< Iz< < _m IIDx P where the constant K depends only on G, m, p, q, a. In particular, for a = O, we have x E L (G) for every q, 1 < q < + oo, with l/q > 1/p - m/v, and |x|ii < K jlxll. 95

Thus, the identity transformation carrying an element x of W (G) into the same function x as an element of L (G) is a p q bounded transformation Wm(G) - L (G). P q (b) Eech derivative Dx, O < Ia! < m-l, is actually continuous on clG, provided 1/p < (m-aI X)/v, and then IDx(t) I< K ll < m |IDBxlpI t E clG, where the constant K depends only on G, m, p, a. In particular, for a = O, x is continuous in clG provided 1/p < m/v, and tx(t)l < K jlx.lm. Thus, the identity transformation carrying an element x of im(G) into the same function x as an element of C is a bounded transformation W (G) - C(clG). p If p > 1, the transformations defined in (a) and (b) are not only bounded, but compact. In other words, under the condition of (a), a = 0, Wm(G) + L (G), and if [xk] is a sequence of functions xk E Wm(G), Ixki < M p q k] p kpthen a suitable subsequence converges (strongly) in L q(G). Under the conditions of (b), c = O, Wm(G) - C(G) and [xk] as above, then the functions xk are equicontinuous in clG. These statements hold even for p = 1, provided the derivatives of maximal order Dxk, Iac = m, k = 1,2,..., are known to be 96

equiabsolutely integrable in G. A more precise form of these results will be given below in (VII ll.ii). (VII ll.ii) If p > 1, G is a region of class K in E, v > 1, m > 1, if [xk] is a sequence of elements xk e Wm(G) with lxkm < M, k = 12,..., for some p kpconstant M, then there is a subsequence [k I and an element x E Wm(G) such s P that: (a) If v > mp, then xk - x strongly in Lq (G) for every q with l/q > 1/p - m/v. (b) If v < mp, then all xk and x are continuous on clG and xks + x uniformly on clG. a a (c) For every 0 < ai < m-l, if v > (m-Ial)p, then D xk D x strongly in L (G) for every 1 < q < + o with l/q > 1/p - (m-IaCl )/v. (d) For every 0 < ai <m-l, if v < (m-Ial)p, then all D xk a a a and D x are continuous on clG, and D xks - D x uniformly on clG. (e) For every Jal = m, D xk s D x weakly in L (G) as s - a. a a Concerning the boundary values 7xk, yxD Xk D x, 0 mp> 1, then 7Xks - yx strongly in nqL (G) for 97

every 1 < q* < + X with 1 < q* < (v-l)p/(v-mp). (The case v < mp is included in (b).) (g) For every 0 < Ial < m-l, if v > (m-Ial)p > 1, then 7D Xks - Dx strongly in Lq (aG) for every 1 < q* < (v-l)p/(v(m- al )p). Statements (abcdefg) above hold even for p = 1 provided we know the partial derivatives D xk of maximal order lat = m are equiabsolutely integrable in G. (h) For p > 1, in all cases it is true that the sequence [k ] a a can be so chosen that xks - x, D Xks - D x strongly in L (G) for all 0 < al < m-l, and D xk D x weakly in p. L (G) for Jai = m. (k) For p > 1, in all cases it is true that the sequence [k ] a a can be so chosen that Yxks -+ x, YD Xks yD x strongly in Lp(G) for all 0 < tal < m-l. Statements (h), (k) hold even for p = 1 provided we know that the partial derivatives D xk of maximal order tat = m are equiabsolutely integrable in G. 98

VII 12. BIBLIOGRAPHICAL NOTES The initial idea of what today are called Sobolev functions can be traced in B. Levi [123] (1906), who used them in problems of the calculus of variations and elliptic differential equations. A more systematic theory of functions x c W1(G) can be seen in G. C. Evans [119] (1920), who used them in potential theory. The absolute continuous functions of L. Tonelli [ ] (1926), or functions A C T, are the functions x c W (G) n C(G), that is, those Sobolev functions x E W1(G) which are continuous in G, or in clG. Tonelli used these functions systematically in two dimensional problems of the calculus of variations (free problems) and in surface area [ ] theory. The full use of Sobolev functions in the calculus of variations and partial differential equations started with C. B. Morrey [73bc] (1934-42), and continued with G. Stampacchia [95abcd], and many others. Meanwhile S. L. Sobolev [94ab] discovered the embedding theorems, and his work was soon continued by V. I. Kondrosov [122]. In the presentation above section VII 6 contains essentially the ideas of G. C. Evans and C. B. Morrey. Concerning the enmbedding theorems we have essentially used the pattern of N. Dunford and J. T. Schwartz [121], proving these theorems first for a half space (section VII 5 above), and then extending to arbitrary open sets (section VII 11) (of Morrey class K here, of 00 class C in N. Dunford and J. T. Schwartz. Nevertheless we have improved the presentation, eliminating a few oversights, and including the weak compactness theorems for p = 1, which have been left out both in S. L. Sobolev as well as in N. Dunford and J. T. Schwartz. The case p = 1 is particularly 99

relevant in the calculus of variations(Tonelli's existence theorems, and some of the present extensions to Lagrange problems.) 100

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