In this paper we prove a new variant of the Herz majorizing principle for operators defined by K-bi-invariant kernels with certain large-scale cancellation properties. As an application, we prove L-P-boundedness of operators defined by Fourier multipliers which satisfy singular differential inequalities of the Hormander-Michlin type. We also find sharp bounds on the L-P-norm of large imaginary powers of the critical L-P-Laplacian. [References: 24]
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