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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.
Chapter 2 covers the foundational inequalities for integrals of functions in Euclidean space. The two key results in this chapter are that symmetric decreasing rearrangement of a continuous function decreases the modulus of continuity, and that certain integral expressions increase when functions are replaced by their symmetric decreasing rearrangement. Other results include the Hardy-Littlewood inequality and the contractivity of rearrangements in the L-infinity norm.
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