The finite Hilbert transform acting in the Zygmund space LlogL

2022/12/17 by Guillermo P. Curbera, Curbera, Guillermo P., Susumu Okada +3
Mathematics · #46E30 #47B34 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Primary 44A15 #Secondary 47A53

paper · pdf · doi:10.48550/arxiv.2212.08835

openalex publication_date 2022/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The finite Hilbert transform T is a singular integral operator which maps the Zygmund space LlogL:=LlogL(-1,1) continuously into L¹:=L¹(-1,1). By extending the Parseval and Poincaré-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation T(f)=g whenever the data function g lies in the range of T within L¹ (shown to contain LlogL). Until now this was only known for g belonging to the union of all L^p spaces with p>1. It is established (due to a result of Stein) that T cannot be extended to any domain space beyond LlogL whilst still taking its values in L¹, i.e., T:LlogL→ L¹ is optimally defined.

Related

• Inversion and extension of the finite Hilbert transform on (-1,1)
• L^p boundedness of the Hilbert transform
• Measure theoretic aspects of the finite Hilbert transform
• Non-extendability of the finite Hilbert transform
• On the Hilbert transform and lacunary directions in the plane