Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators. (1997) - Delaubenfels, Ralph; Lei, Yansong
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 ) .