The Markov–Stieltjes transform on Hardy and Lebesgue spaces

Abstract

We prove that the Markov–Stieltjes transform is a bounded non compact Hankel operator on Hardy space Hᵖ with Hilbert matrix with respect to the standard Schauder basis of Hᵖ and a bounded noncompact operator on Lebesgue space Lᵖ [0, 1] for p ∈ (1,∞) and obtain estimates for its norm in this spaces. It is shown that the Markov–Stieltjes transform on L²(0, 1) is unitary equivalent to the Markov–Stieltjes transform on H². Inverse formulas and operational properties for this transform are obtained.