The method of stationary phase for once integrated group

Abstract

We obtain a formula of decomposition for Φ(A) = A R n S(f (x))ϕ(x) dx + R n ϕ(x) dx using the method of stationary phase. Here (S(t)) t∈R is once integrated, exponentially bounded group of operators in a Banach space X, with generator A, which satisfies the condition: For every x ∈ X there exists δ = δ(x) > 0 such that S(t)x t 1/2+δ → 0 as t → 0. The function ϕ(x) is infinitely differentiable, defined on R n , with values in X, with a compact support supp ϕ, the function f (x) is infinitely differentiable, defined on R n , with values in R, and f (x) on supp ϕ has exactly one nondegenerate stationary point x 0 .