Non-extendability of the finite Hilbert transform

Abstract

We present a detailed survey of recent developments in the study of the finite Hilbert transform and its corresponding inversion problem in rearrangement invariant spaces on (−1, 1). Contents 1. Introduction 1 2. Preliminaries 3 3. The airfoil equation: L p -theory 5 4. The finite Hilbert transform in rearrangement invariant spaces 9 5. Inversion of the finite Hilbert transform and the airfoil equation 11 6. Extension of the finite Hilbert transform 13 7. The finite Hilbert transform acting on LlogL 15 8. The fine spectrum of the finite Hilbert transform 19 9. Integral representation of the finite Hilbert transform 25 References 30