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

Ralph Delaubenfels and Yansong Lei

Abstract and Applied Analysis, 1997, vol. 2, 1-16

Abstract:

Let i A j ( 1 ≤ j ≤ n ) be generators of commuting bounded strongly continuous groups, A ≡ ( A 1 , A 2 , … , 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 B C k ( f ( R n ) ) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when f ( R n ) ⫅ R , then, for appropriate k , r , t ↦ ( 1 − i t ) − k e − i t f ( A ) ( 1 + | A | 2 ) − r is a Fourier-Stieltjes transform, and when f ( R n ) ⫅ [ 0 , ∞ ) , then t ↦ ( 1 + t ) − k e − t f ( A ) ( 1 + | A | 2 ) − r is a Laplace-Stieltjes transform. With A ≡ i ( D 1 , … , D n ) , f ( A ) is a pseudodifferential operator on L p ( R n ) ( 1 ≤ p < ∞ ) or B U C ( R n ) .

Date: 1997
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:472650

DOI: 10.1155/S1085337597000304

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