Book contents
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- 1 The Absolute Convergence Problem
- 2 Holomorphic Functions on Polydiscs
- 3 Bohr’s Vision
- 4 Solution to the Problem
- 5 The Fourier Analysis Point of View
- 6 Inequalities I
- 7 Probabilistic Tools I
- 8 Multidimensional Bohr Radii
- 9 Strips under the Microscope
- 10 Monomial Convergence of Holomorphic Functions
- 11 Hardy Spaces of Dirichlet Series
- 12 Bohr’s Problem in Hardy Spaces
- 13 Hardy Spaces and Holomorphy
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
6 - Inequalities I
from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs
Published online by Cambridge University Press: 19 July 2019
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- 1 The Absolute Convergence Problem
- 2 Holomorphic Functions on Polydiscs
- 3 Bohr’s Vision
- 4 Solution to the Problem
- 5 The Fourier Analysis Point of View
- 6 Inequalities I
- 7 Probabilistic Tools I
- 8 Multidimensional Bohr Radii
- 9 Strips under the Microscope
- 10 Monomial Convergence of Holomorphic Functions
- 11 Hardy Spaces of Dirichlet Series
- 12 Bohr’s Problem in Hardy Spaces
- 13 Hardy Spaces and Holomorphy
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
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.
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- Information
- Dirichlet Series and Holomorphic Functions in High Dimensions , pp. 129 - 152Publisher: Cambridge University PressPrint publication year: 2019