Marcinkiewicz-Zygmund inequalities in variable Lebesgue spaces

We study ${\mathrm{\ell }}^r$-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize $1\le r\le \mathrm{\infty }$ such that every bounded linear operator $T\text{\hspace{0.17em}⁣}:L^{q(\cdot )}({\mathrm{\Omega }}_2,\mu )\to L^{p(\cdot )}({\mathrm{\Omega }}_1,\nu )$ has a bounded ${\mathrm{\ell }}^r$-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.