Polynomial estimates on real and complex \(L_p(\mu)\) spaces (Q2833668)

For a Banach space \(X\) and a positive number \(m\), the \(m\)-polarization constant \(c_m(X)\) is the smallest number such that \(\|L\|\leq c_m(X) \|\widehat L\|\), for every continuous symmetric \(m\)-linear form \(L\in\mathcal L^s(^mX)\) (\(\widehat L\) being the \(m\)-homogeneous polynomial associated to \(L\)). Lawrence Harris in his commentary to Problem 73 of the Scottish book [\textit{R. D. Mauldin} (ed.), The Scottish Book. Mathematics from the Scottish Cafe. Boston etc.: Birkhäuser (1981; Zbl 0485.01013)] proposed the following generalization of this concept.NEWLINENEWLINELet \(k_1,\dots , k_n\) be nonnegative integers such that \(k_1+\cdots + k_n=m\), the constant \(c(k_1,\dots, k_n;X)\) is the smallest number such that, for every \(L\in\mathcal L^s(^mX)\), NEWLINE\[NEWLINE \sup_{\|x_i\|\leq 1,\;i=1,\dots, n}|L(x_1^{k_1},\dots, x_n^{k_n})|\leq c(k_1,\dots, k_n;X)\|\widehat L\|. NEWLINE\]NEWLINENEWLINENEWLINE\textit{L. A. Harris} [in: Analyse fonct. Appl., C. R. Colloq. d'Analyse, Rio de Janeiro 1972, 145--163 (1975; Zbl 0315.46040); J. Math. Anal. Appl. 208, No. 2, 476--486 (1997; Zbl 0898.46045)] estimated this constant, both in the simpler setting of complex Banach spaces and in the more complicated framework of real Banach spaces. He also obtained results for the particular case of \(X=\ell_p\) and \(x_1,\dots, x_n\) with disjoint supports.NEWLINENEWLINEThe article under review focuses on the estimation of \(c(k_1,\dots, k_n;X)\) for spaces \(X=L_p(\mu)\) (both complex and real), considering the target set \(x_1,\dots, x_n\) of norm-one vectors with disjoint supports. In the more involved real case, three different approaches are followed: the first one uses the polarization formula, the second one depends on Clarkson's inequality, and the third one derives from Hoeffding's inequality. In the last section, it is described for which values of \(p\), \(m\) and \(k_1,\dots , k_n\), each approach provides better estimations of \(c(k_1,\dots, k_n;L_p(\mu))\).