Generalized Morrey spaces and trace operator

TL;DR

This paper develops the theory of generalized Morrey spaces, focusing on the trace operator, and introduces atomic decompositions to clarify the role of parameters in these spaces.

Contribution

It extends the theory of generalized Morrey spaces, analyzes the trace operator, and establishes atomic decompositions for these spaces.

Findings

Clarified the role of parameters in generalized Morrey spaces

Established the trace property for these spaces

Derived atomic decompositions for generalized Morrey spaces

Abstract

The theory of generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.