Academic Journal
Hilbert points in Hardy spaces
| Title: | Hilbert points in Hardy spaces |
|---|---|
| Authors: | Brevig, Ole Fredrik, Ortega-Cerdà, Joaquim, Seip, Kristian |
| Source: | Articles publicats en revistes (Matemàtiques i Informàtica) Dipòsit Digital de la UB instname |
| Publication Status: | Preprint |
| Publisher Information: | American Mathematical Society (AMS), 2023. |
| Publication Year: | 2023 |
| Subject Terms: | Hardy spaces, Mathematics - Complex Variables, Funcions de variables complexes, Espais de Hardy, 0211 other engineering and technologies, Anàlisi harmònica, 02 engineering and technology, 01 natural sciences, Functions of complex variables, Functional Analysis (math.FA), Harmonic analysis, Mathematics - Functional Analysis, Inequalities (Mathematics), Mathematics - Classical Analysis and ODEs, H-espaces, H-espais, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 0101 mathematics, Complex Variables (math.CV), Desigualtats (Matemàtica) |
| Description: | A Hilbert point in H p ( T d ) H^p(\mathbb {T}^d) , for d ≥ 1 d\geq 1 and 1 ≤ p ≤ ∞ 1\leq p \leq \infty , is a nontrivial function φ \varphi in H p ( T d ) H^p(\mathbb {T}^d) such that ‖ φ ‖ H p ( T d ) ≤ ‖ φ + f ‖ H p ( T d ) \| \varphi \|_{H^p(\mathbb {T}^d)} \leq \|\varphi + f\|_{H^p(\mathbb {T}^d)} whenever f f is in H p ( T d ) H^p(\mathbb {T}^d) and orthogonal to φ \varphi in the usual L 2 L^2 sense. When p ≠ 2 p\neq 2 , φ \varphi is a Hilbert point in H p ( T ) H^p(\mathbb {T}) if and only if φ \varphi is a nonzero multiple of an inner function. An inner function on T d \mathbb {T}^d is a Hilbert point in any of the spaces H p ( T d ) H^p(\mathbb {T}^d) , but there are other Hilbert points as well when d ≥ 2 d\geq 2 . The case of 1 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 > ∞ 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 φ \varphi that is a Hilbert point in H p ( T 3 ) H^p(\mathbb {T}^3) for p = 2 , 4 p=2, 4 , but not for any other p p ; this is verified rigorously for p > 4 p>4 but only numerically for 1 ≤ p > 4 1\leq p>4 . |
| Document Type: | Article |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1547-7371 1061-0022 |
| DOI: | 10.1090/spmj/1760 |
| DOI: | 10.48550/arxiv.2106.07532 |
| Access URL: | http://arxiv.org/abs/2106.07532 https://hdl.handle.net/2445/200873 http://hdl.handle.net/10852/93947 http://hdl.handle.net/2445/200873 |
| Rights: | arXiv Non-Exclusive Distribution URL: https://www.ams.org/publications/copyright-and-permissions |
| Accession Number: | edsair.doi.dedup.....8dfb789408e049356eb4d0daa1b4d4e5 |
| Database: | OpenAIRE |
| ISSN: | 15477371 10610022 |
|---|---|
| DOI: | 10.1090/spmj/1760 |