Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators (Q1809792)

Summary: Let \(iA_j\) \((1\leq j\leq n)\) be generators of commuting bounded strongly continuous groups, \(A\equiv (A_1,A_2,\dots, A_n)\). 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 \(BC^k(f(\mathbb{R}^n))\) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when \(f(\mathbb{R}^n)\subseteq \mathbb{R}\), then, for appropriate \(k,r,t\mapsto(1- it)^{-k} e^{-itf(A)}(1+|A|^2)^{-r}\) is a Fourier-Stieltjes transform, and when \(f(\mathbb{R}^n)\subseteq [0,\infty)\), then \(t\mapsto(1+ t)^{-k}e^{-tf(A)}(1+|A|^2)^{-r}\) is a Laplace-Stieltjes transform. With \(A\equiv i(D_1,\dots, D_n)\), \(f(A)\) is a pseudo-differential operator on \(L^p(\mathbb{R}^n)\) \((1\leq p<\infty)\) or \(BUC(\mathbb{R}^n)\).