Variable exponent Campanato spaces

Abstract

We study variable exponent Campanato spaces on spaces of homogeneous type. We prove an embedding result between variable exponent Campanato spaces. We also prove that these spaces are equivalent, up to norms, to variable exponent Morrey spaces L p(·),λ(·) (X) with λ+ < 1 and variable exponent Holder spaces H α(·)(X) with λ− > 1. In the setting of an arbitrary quasimetric measure spaces, we introduce the log-Holder condition for p(x) with the distance d(x, y) replaced by μB(x, d(x, y)), which provides a weaker restriction on p(x) in the general setting and show that some basic facts for variable exponent Lebesgue spaces hold without the assumption that X is homogeneous or even Ahlfors lower or upper regular. However, the main results for Campanato spaces are proved in the setting of homogeneous spaces X. Bibliography: 34 titles.