Summation of Coefficients of Polynomials on $\ell_{p}$ Spaces

Abstract

We investigate the summability of the coefficients of $m$-homogeneous polynomials and $m$-linear mappings defined on $\ell_{p}$ spaces. In our research we obtain results on the summability of the coefficients of $m$-linear mappings defined on $\ell_{p_{1}} \times \dotsb \times \ell_{p_{m}}$. The first results in this respect go back to Littlewood [17] and Bohnenblust and [6] for bilinear and $m$-linear forms on $c_{0}$, and Hardy and Littlewood [15] and Praciano-Pereira [20] for bilinear and $m$-linear forms on arbitrary $\ell_{p}$ spaces. Our results recover and in some case complete these old results through a general approach on vector valued $m$-linear mappings.