Density, Duality and Preduality in Grand Variable Exponent Lebesgue and Morrey Spaces

Abstract

In this note some structural properties of grand variable exponent Lebesgue/Morrey spaces over spaces of homogeneous type are obtained. In particular, it is proved that the closure of $$L^{\infty }$$ and the closure of $$L^{p(\cdot ), \lambda (\cdot )}$$ in grand variable exponent Morrey space $$L^{p(\cdot ), \lambda (\cdot ), \theta }$$ coincide if the measure of the underlying space is finite. Moreover, we get two different characterizations of this class. Further, duality and preduality of grand variable exponent Lebesgue spaces defined on quasi-metric measure spaces with $$\sigma $$ -finite measure are obtained, which is new even when $$\mu $$ is finite.