On Hankel conjugate functions

Abstract

We consider some aspects of harmonic analysis of the differential operator . Spectral decomposition of its self-adjoint extension is given in terms of the Hankel transform Hν. We present a fairly detailed analysis of the corresponding Poisson semigroup {Pt}t > 0: this is given in a weighted setting with Ap-weights involved. Then, we consider conjugate Poisson integrals of functions from Lp(w), w ∈ Ap, 1 ≦ p < ∞. Boundary values of the conjugate Poisson integrals exist both in Lp(w) and a.e., and the resulting mapping is called the generalized Hilbert transform. Mapping properties of that transform are then proved. All this complements, in some sense, the analysis of conjugacy for the modified Hankel transform Hν which was initiated in the classic paper of Muckenhoupt and Stein [15], then continued in a series of papers by Andersen, Kerman, Rooney and others.