Marcinkiewicz–Zygmund inequalities in variable Lebesgue spaces

Abstract

We study ℓ^r-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize 1 ≤ r ≤ ∞ such that every bounded linear operator T : L^q(·)(Ω_2, µ) →L^p(·)(Ω_1, ν) has a bounded ℓ^r-valued extension. We consider both non-atomic measures and measures with atoms and show the differences that can arise. We present some applications of our results to weighted norm inequalities of linear operators and vector-valued extensions of fractional operators with rough kernel.