Polynomial estimates on real and complex spaces
Tom 235 / 2016
Streszczenie
In his commentary to Problem 73 of Mazur and Orlicz in the Scottish Book, L. A. Harris raised the following natural generalization: Let X be a Banach space, let k_1,\ldots,k_n be nonnegative integers whose sum is m and let c(k_1, \ldots, k_n; X) be the smallest number with the property that if L is any symmetric m-linear mapping of one real normed linear space into another, then |L(x_1^{k_1}\ldots x_n^{k_n})|\leq c(k_1,\ldots,k_n; X)\|\widehat L\|, where \widehat L is the m-homogeneous polynomial associated to L. In this paper, we give estimates in the case of a real L_p(\mu) space using three different techniques and we get optimal results in some special cases.