6 - Inequalities I

Summary

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.