Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators. (1997)

• Record Type: Journal Article
• Authors: Delaubenfels, Ralph; Lei, Yansong
• Is Part Of: Abstract and applied analysis. Volume 2:Number 1/2(1997)
• Page Start: 121
• Page End: 136
• Publication Date: 1997
• Subjects: Regularized functional calculi -- semigroups -- cosine functions -- pseudodifferential operators; Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/S1085337597000304
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : Leti A j ( 1 ≤ j ≤ n ) be generators of commuting bounded strongly continuous groups, A ≡ ( A 1, A 2, …, A n ) . We show that, whenf has sufficiently many polynomially bounded derivatives, then there existk, r > 0 such thatf ( A ) has a( 1 + | A | 2 ) − r -regularizedB C k ( f ( R n ) ) functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, whenf ( R n ) ⫅ R, then, for appropriatek, r, t ↦ ( 1 − i t ) − k e − i t f ( A ) ( 1 + | A | 2 ) − r is a Fourier-Stieltjes transform, and whenf ( R n ) ⫅ [ 0, ∞ ), thent ↦ ( 1 + t ) − k e − t f ( A ) ( 1 + | A | 2 ) − r is a Laplace-Stieltjes transform. WithA ≡ i ( D 1, …, D n ), f ( A ) is a pseudodifferential operator onL p ( R n ) ( 1 ≤ p < ∞ ) orB U C ( R n ) .