Measure theoretic aspects of the finite Hilbert transform

2024/06/23 by Curbera, Guillermo P., Okada, Susumu, Ricker, Werner J.
#28B05 #44A15 #46E30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2406.16233

Abstract

The finite Hilbert transform T, when acting in the classical Zygmund space \logl (over (-1,1)), was intensively studied in \citecurbera-okada-ricker-log. In this note an integral representation of T is established via the L¹(-1,1)-valued measure \mlog\colon A↦ T(χ_A) for each Borel set A⊆(-1,1). This integral representation, together with various non-trivial properties of \mlog, allow the use of measure theoretic methods (not available in \citecurbera-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.

Related

• Non-extendability of the finite Hilbert transform
• Fine spectra of the finite Hilbert transform in function spaces
• The finite Hilbert transform acting in the Zygmund space LlogL
• The Hilbert Transform of a Measure
• Inversion and extension of the finite Hilbert transform on (-1,1)