Variable exponent Campanato spaces

We study variable exponent Campanato spaces L^p(·),λ(·)(X) 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 Hölder spaces H^α(·)(X) with λ_- > 1. In the setting of an arbitrary quasimetric measure spaces, we introduce the log-Hölder 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.