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2 results

  • 18 - Hardy–Littlewood Inequality

  • from Part 3 - Replacing Polydiscs by Other Balls
  • Andreas Defant, Carl V. Ossietzky Universität Oldenburg, Germany, Domingo García, Universitat de València, Spain, Manuel Maestre, Universitat de València, Spain, Pablo Sevilla-Peris
  • Book:
    Dirichlet Series and Holomorphic Functions in High Dimensions
    Published online:
    19 July 2019
    Print publication:
    08 August 2019, pp 475-485

  • Summary

    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.

  • 2 - Main Inequalities on Rn3

  • Albert Baernstein II, Washington University, St Louis
  • Book:
    Symmetrization in Analysis
    Published online:
    22 February 2019
    Print publication:
    14 March 2019, pp 54-91

  • Summary

    Chapter2 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 inequalityand the contractivity of rearrangements in the L-infinity norm.