Regularized functional calculi, semigroups, and cosine functionsfor pseudodifferential operators
Ralph deLaubenfels, Yansong Lei · Abstract and Applied Analysis · 1997
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 ‐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, is a Fourier‐Stieltjes transform, and when f(Rn)⫅[0, ∞), then is a Laplace‐Stieltjes transform. With A ≡ i(D1, …, Dn), f(A) is a pseudodifferential operator on Lp(Rn)(1 ≤ p < ∞) or BUC(Rn).