Applications of the theory of imaginary powers of operators
Michael J. Fisher · Rocky Mountain Journal of Mathematics · 1972
Imaginary powers of directional derivatives and of certain other operators are used to study semigroups which arise in the analysis of singular integral operators.Imaginary powers of directional derivatives are used to estimate the maximal functions and the Littlewood-Paley g-function of the Poisson integral on a Hilbert space. I. Introduction.The purpose of this paper is to study some of the implications of the existence as bounded operators of purely imaginary powers of the infinitesimal generators of certain semigroups.The setting of the paper will be Classical Analysis on Hilbert Space.Let H be a real separable Hilbert space and let L P (H) denote the Banach space of p-power integrable functions with respect to the normal distribution with variance parameter 1.Let y->T y denote the regular representation of the additive group of H as isometries on Lp(H).Fix p in 1 0, and if ( -T) denotes the infinitesimal generator of P z , z > 0, then (D h ) ic and T ic are strongly continuous groups of