Optimal Regularity Properties of the Generalized Sobolev Spaces

We prove optimal embeddings of the generalized Sobolev spaces , where is a rearrangement invariant function space, into the generalized H？lder-Zygmund space generated by a function space . 1. Introduction The classical Sobolev space , , consists of all locally integrable functions , defined on , , with the Lebesgue measure, such that the following norm is finite: , where stands for the -norm. In investigating the regularity of the function , we may assume, without any loss of generality, that , is a domain in , and is zero outside . For simplicity we suppose that the Lebesgue measure of equals one and that the origin lies in . It is well known that in the supercritical case , where , is the H？lder-Zygmund space (see [1]). In the critical case the function may not be even continuous. The result (1) is not optimal. We prove that the optimal one is obtained if in (1) is replaced by the Marcinkiewicz space . In this paper we prove similar optimal results, when is replaced by a more general rearrangement invariant space . The Sobolev space consists of all with a finite quasinorm . More precisely, we consider quasinormed rearrangement invariant spaces , consisting of functions , such that the quasinorm , where is a monotone quasinorm, defined on with values in and is the cone of all locally integrable functions on with the Lebesgue measure. Monotonicity means that implies . We suppose that , which means continuous embeddings. Here is the decreasing rearrangement of , given by , and is the distribution function of , defined by denoting Lebesgue -measure. Note that for . Finally, . Let , be the Boyd indices of . For example, if , then and the condition means . Note that for this is always satisfied. For these reasons we suppose that for the general , and the case is called super-critical, while the case -critical. In the super-critical case the function is always continuous, while the spaces in the critical case , can be divided into two subclasses: in the first subclass the functions may not be continuous—then the target space is rearrangement invariant, while these functions in the second subclass are continuous and the target space is the generalized H？lder-Zygmund space (see Definition 1). The separating space for these two subclasses is given by the Lorentz space , . If ; then consists of continuous functions (see the classical result of Stein [2]). The main goal of this paper is to prove optimal embeddings of the Sobolev space into the generalized H？lder-Zygmund space . First we prove that this embedding for is equivalent to the continuity of the operator