Marcinkiewicz–Zygmund inequalities in variable Lebesgue spaces

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