An embedding theorem for Sobolev type functions with gradients in a Lorentz space

Abstract

Abstract. The purpose of this paper is to prove an embedding theorem for Sobolev type functions whose gradients are in a Lorentz space, in the framework of abstract metricmeasure spaces. We then apply this theorem to prove absolute continuity and differentiability of such functions.Introduction. In this article, we extend Morrey's embedding theorem (see for instance [GT, Thm. 7.17]) to Sobolev type functions whose generalized gradients are in a Lorentz space, when the underlying space is a metric-measure space. In fact, Morrey's theorem states that for any function f in the Sobolev space W 1,p (R n ), the following inequality is satisfied:

Space filling with metric measure spaces

“…As the conclusion of the theorem is invariant under bi-Lipschitz mappings, we may assume that X is geodesic. Hence the proof of [21,Proposition 1.4] shows that the hypotheses of [21, Theorem 2.1] are met under our assumptions. As before, we let C be a number, possibly varying at each instance, that depends only on the constants associated to our assumptions.…”

“…In the Euclidean setting, this was done in [13]. In the metric setting, closely related results have been given in [22] and [21]. Our proof follows the outline of [13], and hence we occasionally skip a few details.…”

“…This principle has been recently extended to the abstract metric setting [21,22]. The crucial property for this paper is Lusin's condition N.…”

“…More precise versions of this result can likely be deduced from [21]. (X, d, μ) is complete, doubling, supports a Q-Poincaré inequality, and is Q-regular at small scales.…”

Space-filling vs. Luzin's condition (N)

“…We use the symbol to indicate that the mapping under consideration is a surjection. 2 These spaces have also been studied in[RM09] and[Rom08] in the metric setting and in[KKM99] in the Euclidean setting.…”