Plancherel –Type Estimates and Operator-Valued Fourier Multipliers of and Sobolev Spaces

Abstract

We show basic characterizations of Hilbert spaces and the vector –valued Littlewood-Paley and Fourier multipliers theorems. We also show the operator-valued Fourier multiplier theorem and maximal regularity .We study the behavior of inner functions and summability kernels for the Lebesgue space multipliers. We investigate the multipliers in Hardy-Sobolev spaces and show remarks on vector-valued BMOA and vector-valued multipliers. We develop a very general operator-valued functional calculus for operators with an -calculus .We study the operator –valued Fourier multiplier theorem on the Lebesgue space and geometry of Banach spaces. We give the Gaussian estimates and the -boundedness of Riesz means. The Plancerel-type estimates and sharp spectral multipliers are considered.