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

Ralph Delaubenfels and Yansong Lei

Abstract and Applied Analysis, 1997, vol. 2, issue 1-2, 121-136

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

Date: 1997
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2:y:1997:i:1-2:p:121-136

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