Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces

Abstract

The nite Hilbert transform T is a classical (singular) kernel operator which is continuous in every rearrangement invariant space X over (-1,1)  having non-trivial Boyd indices. For X L^p, 1< p < ∞, this operator has been intensively investigated since the 1940's (also under the guise of the "airfoil equation"). Recently, the extension and inversion of T : X →X for more general X has been studied in [6], where it is shown that there exists a larger space rT;Xs, optimal in a well dened sense, which contains X continuously and such that T can be extended to a continuous linear operator[ T : T;X ]→ X. The purpose of this paper is to continue this investigation of T via a consideration of the X-valued vector measure m_X : A →T(X_A) induced by T and its associated integration operator f→ ş1_-1 f dm_X. In particular, we present integral representations of T : X → X based on the L¹-space of m_X and other related spaces of integrable functions.

Key words: Finite Hilbert transform, rearrangement invariant space, optimal domain, vector measure, integral representation.