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Title: | Hilbert points in Hardy spaces |
Author: | Fredrik Brevig, Ole Ortega Cerdà, Joaquim Seip, Kristian |
Keywords: | Espais de Hardy Funcions de variables complexes H-espais Anàlisi harmònica Desigualtats (Matemàtica) Hardy spaces Functions of complex variables H-espaces Harmonic analysis Inequalities (Mathematics) |
Issue Date: | 7-Jun-2023 |
Publisher: | American Mathematical Society (AMS) |
Abstract: | A Hilbert point in $H^p\left(\mathbb{T}^d\right)$, for $d \geq 1$ and $1 \leq p \leq \infty$, is a nontrivial function $\varphi$ in $H^p\left(\mathbb{T}^d\right)$ such that $\|\varphi\|_{H^p\left(\mathbb{T}^d\right)} \leq\|\varphi+f\|_{H^p\left(\mathbb{T}^d\right)}$ whenever $f$ is in $H^p\left(\mathbb{T}^d\right)$ and orthogonal to $\varphi$ in the usual $L^2$ sense. When $p \neq 2, \varphi$ is a Hilbert point in $H^p(\mathbb{T})$ if and only if $\varphi$ is a nonzero multiple of an inner function. An inner function on $\mathbb{T}^d$ is a Hilbert point in any of the spaces $H^p\left(\mathrm{~T}^d\right)$, but there are other Hilbert points as well when $d \geq 2$. The case of 1 -homogeneous polynomials is studied in depth and, as a byproduct, a new proof is given for the sharp Khinchin inequality for Steinhaus variables in the range $2<p<\infty$. Briefly, the dynamics of a certain nonlinear projection operator is treated. This operator characterizes Hilbert points as its fixed points. An example is exhibited of a function $\varphi$ that is a Hilbert point in $H^p\left(\mathbb{T}^3\right)$ for $p=2,4$, but not for any other $p$; this is verified rigorously for $p>4$ but only numerically for $1 \leq p<4$. |
Note: | Versió postprint del document publicat a: https://doi.org/10.1090/spmj/1760 |
It is part of: | St Petersburg Mathematical Journal, 2023, vol. 34, num. 3, p. 405-425 |
URI: | http://hdl.handle.net/2445/200873 |
Related resource: | https://doi.org/10.1090/spmj/1760 |
ISSN: | 1061-0022 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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