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

Eiichi Nakai¹

¹ Department of Mathematics Osaka Kyoiku University Kashiwara, Osaka 582-8582, Japan

Studia Mathematica, Tome 176 (2006) no. 1, pp. 1-19

Résumé

We investigate the relations between the Campanato, Morrey and Hölder spaces on spaces of homogeneous type and extend the results of Campanato, Mayers, and Macías and Segovia. The results are new even for the $\mathbb R^n$ case. Let $(X,d,\mu)$ be a space of homogeneous type and $(X,\delta,\mu)$ its normalized space in the sense of Macías 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 Hölder 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.

Détail

DOI : 10.4064/sm176-1-1

Keywords: investigate relations between campanato morrey lder spaces spaces homogeneous type extend results campanato mayers mac segovia results even mathbb space homogeneous type delta its normalized space sense mac segovia study relations these function spaces delta using these relations theorems campanato morrey lder spaces normal space valid function spaces space homogeneous type application obtain boundedness operators related partial differential equations boundedness fractional differential integral operators characterizations pointwise multipliers