Reduction theorems for Sobolev embeddings into the spaces of Holder, Morrey and Campanato type

Abstract

Let X be a rearrangement-invariant Banach function space on Q where Q is a cube in R-n and let V-1 X( Q) be the Sobolev space of real-valued weakly differentiable functions f satisfying vertical bar del f vertical bar is an element of X(Q). We establish a reduction theorem for an embedding of the Sobolev space V-1 X( Q) into spaces of Campanato, Money and Holder type.

As a result we obtain a new characterization of such embeddings in terms of boundedness of a certain one-dimensional integral operator on representation spaces. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Keywords

• Rearrangement-invariant function spaces
• reduction operator
• Sobolev embeddings
• generalized Campanato
• Money and Holder spaces
• Polya-Szego principle