Singular integrals on symmetric spaces of real rank one

Abstract

In this paper we prove a new variant of the Herz majorizing principle for operators defined by $\mathbb {K}$-bi-invariant kernels with certain large-scale cancellation properties. As an application, we prove $L\sp p$-boundedness of operators defined by Fourier multipliers which satisfy singular differential inequalities of the Hörmander-Michlin type. We also find sharp bounds on the $L\sp p$-norm of large imaginary powers of the critical $L\sp p$-Laplacian.