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 ${\ell }_{p_1}\times \cdots \times {\ell }_{p_m}$. The first results in this respect go back to Littlewood and Bohnenblust and Hille (for bilinear and $m$-linear forms on $c_0$) and Hardy and Littlewood and Praciano-Pereira (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.