The Campanato, Morrey and Holder spaces on spaces of homogeneous type

We investigate the relations between the Campanato, Morrey and Holder spaces on spaces of homogeneous type and extend the results of Campanato, Mayers, and Macias and Segovia. The results are new even for the R-n case. Let (X, d, mu) be a space of homogeneous type and (X, delta, mu) its normalized space in the sense of Macias and Segovia. We also study the relations of these function spaces for (X, d, mu) and for (X, delta, mu). Using these relations, we can show that theorems for the Campanato, Morrey or Holder spaces on the normal space are valid for the function spaces on any space of homogeneous type. As an application we obtain boundedness of some operators related to partial differential equations, boundedness of fractional differential and integral operators, and give characterizations of pointwise multipliers.