EMBEDDINGS AND RELATED TOPICS IN GRAND VARIABLE EXPONENT HAJŁASZ-MORREY-SOBOLEV SPACES

Embeddings in the framework of grand variable exponent function spaces are studied. In particular, embeddings from grand variable exponent Hajlasz-Sobolev-Morrey spaces to variable exponent H center dot older spaces are established. The regularity of a fractional integral operator defined with respect to a non-doubling measure is also investigated. In particular, mapping properties of this operator from a grand variable exponent Morrey space to a grand variable parameter H center dot older space are studied. The results are proved under the log-H center dot older continuity condition on the exponents. The spaces are defined, generally speaking, on quasi-metric measure spaces, however, the results are new even for Euclidean spaces.