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arXiv · 2401.03211

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

Arash Ghorbanalizadeh·Reza Roohi Seraji
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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)}}) $.

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Arash Ghorbanalizadeh, Reza Roohi Seraji. 2024-01-06. Some Remarks on the Vector-Valued Variable Exponent Lebesgue Spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$. https://arxiv.org/abs/2401.03211

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