Book contents
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- 18 Hardy–Littlewood Inequality
- 19 Bohr Radii in ℓp Spaces and Unconditionality
- 20 Monomial Convergence in Banach Sequence Spaces
- 21 Dineen’s Problem
- 22 Back to Bohr Radii
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
18 - Hardy–Littlewood Inequality
from Part 3 - Replacing Polydiscs by Other Balls
Published online by Cambridge University Press: 19 July 2019
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- 18 Hardy–Littlewood Inequality
- 19 Bohr Radii in ℓp Spaces and Unconditionality
- 20 Monomial Convergence in Banach Sequence Spaces
- 21 Dineen’s Problem
- 22 Back to Bohr Radii
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
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.
- Type
- Chapter
- Information
- Dirichlet Series and Holomorphic Functions in High Dimensions , pp. 475 - 485Publisher: Cambridge University PressPrint publication year: 2019