Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators

Abstract

Let iAj(1≤j≤n) be generators of commuting bounded strongly continuous groups, A≡(A1,A2,…,An). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k,r>0 such that f(A) has a (1+|A|2)−r-regularized BCk(f(Rn)) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when f(Rn)⫅R, then, for appropriate k,r, t↦(1−it)−ke−itf(A)(1+|A|2)−r is a Fourier-Stieltjes transform, and when f(Rn)⫅[0,∞), then t↦(1+t)−ke−tf(A)(1+|A|2)−r is a Laplace-Stieltjes transform. With A≡i(D1,…,Dn),f(A) is a pseudodifferential operator on Lp(Rn)(1≤p<∞) or BUC(Rn).