The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive

By Andreas Defant, Leonhard Frerick, J. Ortega‐Cerdà, M. Ounaies, Kristian Seip. 2009

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The Bohnenblust-Hille inequality says that the ‘ 2m m+1 -norm of the coefcients of an m-homogeneous polynomial P on C n is bounded by kPk1 times a constant independent of n, wherekk 1 denotes the supremum norm on the polydisc D n . The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be C m for some C > 1. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc D n behaves asymptotically as p (logn)=n modulo a factor bounded away from 0 and innity,

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2fa41f047637d7414f037cab9051bcbadbdeece7

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Citations in the corpus

Lower bounds for the constants in the Bohnenblust–Hille inequality: The case of real scalars (2011, background): “The constants of the polynomial version of the Bohnenblust-Hille inequality (complex case) were recently investigated in [3].”
The Bohr radius for starlike logharmonic mappings (2016, background): “Bohr radius and local Banach space theory,[13,14] and obtained the optimal asymptotic estimate for the n-dimensional Bohr radius on the polydisk U n .[15] The Bohr inequality has also emerged as an active area of research for operator algebraists after Dixon [16] used it to settle in the negative a ”
Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables (2014, methodology): “37 1 Introduction Recall from [11] that the precise asymptotic order of the Sidon constant of all finite Dirichlet polynomials ∑N n=1 ann −s is given by sup a1,.”
Multidimensional analogues of refined Bohr’s inequality (2020, background): “Several other aspects and generalizations of Bohr’s inequality may be obtained from [8, 10, 18, 21, 27, 32, 33, 34, 35], the monograph of Kresin and Maz’ya [28], and the references therein.”
Bohr radius for subordinating families of analytic functions and bounded harmonic mappings (2014, background): “[11] obtained the optimal asymptotic estimate for the n-dimensional Bohr radius on the polydisk U.”