Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.16233 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913402558873600 |
|---|---|
| author | Curbera, Guillermo P. Okada, Susumu Ricker, Werner J. |
| author_facet | Curbera, Guillermo P. Okada, Susumu Ricker, Werner J. |
| contents | The finite Hilbert transform $T$, when acting in the classical Zygmund space $\logl$ (over $(-1,1)$), was intensively studied in \cite{curbera-okada-ricker-log}. In this note an integral representation of $T$ is established via the $L^1(-1,1)$-valued measure $\mlog\colon A\mapsto T(χ_A)$ for each Borel set $A\subseteq(-1,1)$. This integral representation, together with various non-trivial properties of $\mlog$, allow the use of measure theoretic methods (not available in \cite{curbera-okada-ricker-log}) to establish new properties of $T$. For instance, as an operator between Banach function spaces $T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for $T$ plays a crucial role. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measure theoretic aspects of the finite Hilbert transform Curbera, Guillermo P. Okada, Susumu Ricker, Werner J. Functional Analysis 44A15, 28B05, 46E30 The finite Hilbert transform $T$, when acting in the classical Zygmund space $\logl$ (over $(-1,1)$), was intensively studied in \cite{curbera-okada-ricker-log}. In this note an integral representation of $T$ is established via the $L^1(-1,1)$-valued measure $\mlog\colon A\mapsto T(χ_A)$ for each Borel set $A\subseteq(-1,1)$. This integral representation, together with various non-trivial properties of $\mlog$, allow the use of measure theoretic methods (not available in \cite{curbera-okada-ricker-log}) to establish new properties of $T$. For instance, as an operator between Banach function spaces $T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for $T$ plays a crucial role. |
| title | Measure theoretic aspects of the finite Hilbert transform |
| topic | Functional Analysis 44A15, 28B05, 46E30 |
| url | https://arxiv.org/abs/2406.16233 |