Abstract and Applied Analysis

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

Ralph Delaubenfels and Yansong Lei

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Let iAj(1jn) 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(1it)keitf(A)(1+|A|2)r is a Fourier-Stieltjes transform, and when f(Rn)[0,), then t(1+t)ketf(A)(1+|A|2)r is a Laplace-Stieltjes transform. With Ai(D1,,Dn),f(A) is a pseudodifferential operator on Lp(Rn)(1p<) or BUC(Rn).

Article information

Abstr. Appl. Anal., Volume 2, Number 1-2 (1997), 121-136.

First available in Project Euclid: 7 April 2003

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Zentralblatt MATH identifier

Primary: 47A60: Functional calculus
Secondary: 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47D09: Operator sine and cosine functions and higher-order Cauchy problems [See also 34G10] 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Regularized functional calculi semigroups cosine functions pseudodifferential operators


Delaubenfels, Ralph; Lei, Yansong. Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators. Abstr. Appl. Anal. 2 (1997), no. 1-2, 121--136. doi:10.1155/S1085337597000304. https://projecteuclid.org/euclid.aaa/1049737246

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