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2006.
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 +∫Rnφ (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 → 0 as t → 0. The function φ(x) is infinitely differentiable, defined on Rn, with values in X, with a compact support supp φ, the function f(x) is infinitely differentiable defined on Rn, with values in R, and f(x) on supp φ has exactly one nondegenerate stationary point x0.