Some Remarks on the Vector-Valued Variable Exponent Lebesgue Spaces ${\ell }^{q(\cdot )}(L^{p(\cdot )})$

Abstract

In this paper, we investigate the geometric properties of the variable mixed Lebesgue-sequence space ${\ell }^{q(\cdot )}(L^{p(\cdot )})$ as a Banach space. We show that, if $1<q_{-},p_{-},q_{+},p_{+}<\infty$, then ${\ell }^{q(\cdot )}(L^{p(\cdot )})$ is strictly and uniformly convex. We also prove that when $1\le q_{-},p_{-},q_{+},p_{+}<\infty ,$ the convergence in norm implies the convergence in measure, and under some conditions on exponents, the approximation identity holds in the space ${\ell }^1(L^{\frac{p(\cdot )}{q(\cdot )}})$.

Ready to publish