25 - Inequalities IV

Summary

We look for inequalities that relate some p-norm of the coefficients of a vector-valued polynomial in n variables with a constant (that depends on the degree but not on n) and the supremum of the polynomial on the n-dimensional polydisc (or other n-dimensional balls) . This is an analogue of the Bohnenblust-Hille (and the Hardy-Littlewood inequalities) for vector-valued polynomials that have been extensively studied. This leads in a natural way to cotype. It is shown that if the polynomial takes values in a Banach space with cotype q, then such an inequality is satisfied with the q-norm of the coefficients. The constant that appears grows too fast on the degree. If we want to have a better asymptotic behaviour of the constants a finer property on the space is needed: hypercontractive polynomial cotype. Conditions are given for a space to enjoy this property. A polynomial version of the Kahane inequality is given (all L_p norms are equivalent for polynomials). Finally, these type of inequalities is extended to operators between Banach spaces, leading to the definition of polynomially summing operators, an extension of the classical concept of summing operator.