The method of stationary phase for once integrated group

We obtain a formula of decomposition for Φ(A )= A Rn S(f (x))ϕ(x) dx + Rn 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 t1/2+δ → 0a st → 0. The function ϕ(x) is infinitely differentiable, defined on Rn ,w ith values in X, with a compact support supp ϕ, the function f (x) is infinitely differen- tiable, defined on Rn ,w ith values inR ,a ndf (x) on supp ϕ has exactly one nondegenerate stationary point x0.