Newtonian spaces: An extension of Sobolev spaces to metric measure spaces.

Abstract

This thesis studies a definition of Sobolev spaces on metric measure spaces, using the notion of upper gradients. Let <italic>X</italic> be a space equipped with a metric <italic>d</italic> and a measure mu. If <italic> f</italic> is a real-valued function on <italic>X</italic>, then a non-negative Borel measurable function rho on <italic>X</italic> is said to be an upper gradient of <italic>f</italic> if for each rectifiable curve gamma in <italic> X</italic> the following inequality <display-math> <fd> &vbm0;fx</fen>-f y</fen>&vbm0;&le;g rds </fd> </display-math> holds, where <italic>x</italic> and <italic>y</italic> denote the end-points of gamma. The definition of Sobolev spaces studied in this thesis is as follows. Let 1 &le; <italic>p</italic> < infinity. The space of all <italic>p</italic>-integrable real-valued functions on <italic>X</italic> with <italic>p</italic>-integrable upper gradients is called the Newtonian space of index <italic>p</italic>, in recognition of the fact that the above inequality defining upper gradients is a generalization of the fundamental theorem of calculus. It is proven in this thesis that even with no assumptions on <italic>X</italic>, the Newtonian spaces are Banach spaces. It is shown that for Euclidean domains the Newtonian spaces are indeed the classical Sobolev spaces. It is also shown that real-valued functions on <italic>X</italic> that satisfy a Sobolev type definition of Hajlasz with index <italic> p</italic> are also in the Newtonian space of index <italic>p</italic> for all finite <italic>p</italic> &ge; 1. Furthermore, it is shown that if <italic> X</italic> supports a (1, <italic>q</italic>)-Poincare inequality with 1 &le; <italic>q</italic> < <italic>p</italic> and the measure on <italic> X</italic> is doubling, then functions in the Newtonian space of index <italic> p</italic> satisfy the Hajlasz definition. Other influences of (1, <italic> p</italic>)-Poincare inequality are also explored. It is shown that if <italic>X</italic> supports a (1, <italic>p</italic>)-Poincare inequality and the measure on <italic>X</italic> satisfies some geometric conditions, then Lipschitz functions are dense in the Newtonian space of index <italic> p</italic>, and furthermore, functions in this Newtonian space are continuous outside of sets of arbitrarily small <italic>p</italic>-capacity. Sobolev embedding theorems are also proven under assumptions of Poincare inequalities and some geometric conditions on the measure. This thesis also studies a criterion, called the <italic>MEC_p</italic> criterion, under which the Newtonian space of index <italic>p</italic> is not degenerate; that is, it is strictly smaller than the <italic>L<super> p</super></italic> class. The effect of Poincare inequalities on this criterion is studied, and some examples are explored in this context. The effect of the criterion on <italic>p</italic>-capacity estimates are also studied. For domains in <italic>X</italic>, Newtonian spaces with zero boundary values outside the domain are defined, and it is shown that under certain conditions on <italic>X</italic>, Lipschitz functions are dense in these spaces. In such spaces, the existence of energy-minimizing functions is proved; that is, a Dirichlet-type problem is solved. The main tool used in this thesis is the concept of <italic>p</italic>-modulus of curve families in <italic>X</italic>. This concept was first developed by Fuglede and Ziemer.