Boundedness of the Hilbert Transform in Lorentz Spaces and Applications to Operator Ideals

Abstract

In this paper, it is investigated the optimal range of the classical Hilbert transform in Lorentz spaces of functions and its non-commutative counterparts  including a triangular truncation operator in Schatten-Lorentz ideals. Some applications of obtained results to operator Lipschitz functions and  commutator estimates in Schatten-Lorentz ideals of compact operators are presented.