Marcinkiewicz–Zygmund inequalities in variable Lebesgue spaces

We study \(\ell ^r\)-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize \(1\le r\le \infty \) such that every bounded linear operator \(T:L^{q(\cdot )}(\Omega _2, \mu )\rightarrow L^{p(\cdot )}(\Omega _1, \nu )\) has a bounded \(\ell ^r\)-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.