Littlewood’s and Bohnenblust-Hille’s inequalities (recall Chapter 6) bound certain sequence norms of the coefficients of a polynomial by a constant (not depending on the number of variables) times the supremum of the polynomial on the polydisc. A similar problem is handled here, replacing the polydisc by the unit ball of C^n with some p-norm. Optimal exponents (that depend on the degree of the polynomial and on p) are given. The proof relies on the interplay between homogeneous polynomials and multilinear mappings and an analogous inequality for multilinear mappings. This one is proved by giving a generalized mixed inequality that bounds a mixed norm of the coefficients of a matrix by the supremum on the p-balls of the associated multilinear mapping.

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.

The Bohnenblust-Hille inequality bounds the (2m)/(m+1)-norm of the coefficients of an m-homogeneous polynomial in n variables by a constant (depending on m but not on n) multiplied by the norm (the supremum on the n-dimensional polydisc) of the polynomial. This follows from the inequality for m-linear forms. Littlewood’s inequality shows that the 4/3-norm of a bilinear form is bounded by a constant (not depending on n) multiplied by the norm of the form and that 4/3 cannot be improved. A tool is the Khinchin-Steinhaus inequality, showing that the L_p-norms (for 1 ≤ p < ∞) of a polynomial are equivalent to the l_2 norm of the coefficients. Other tools are inequalities relating mixed norms of the coefficients of a matrix with the norm of the associated multilinear form. All these give the multilinear Bohnenblust-Hille inequality, showing also that the (2m)/(m+1) cannot be improved. The exponent in the polynomial inequality is also optimal (this does not follow from the multilinear case). As a consequence of the inequality we have S^m=(2m)/(m-1) (see Chapter 4). By a generalized Hölder inequality the constant in the multilinear inequality grows at most polynomially on m.