Hilbert transform and Littlewood–Paley theory

Abstract

In this final chapter, we connect the probabilistic notion of UMD spaces with various norm inequalities of harmonic analysis. We begin with the equivalence of the UMD property of X with the boundedness of the classical Hilbert transform on L ^p(ℝ;X) and proceed to more general vector-valued Fourier multiplier operators and Littlewood–Paley inequalities, including versions in several variables and in the periodic setting. Applications of the general theory are illustrated by two independent sections dealing with operators on Schatten classes, and interpolation of Sobolev spaces.