THE METHOD OF STATIONARY PHASE FOR ONCE INTEGRATED GROUP

Ramiz Vugdalic and Fikret Vajzovic

Abstract: We obtain a formula of decomposition for $$ \Phi(A)=A\int\limits_{R^n}{S(f(x))\varphi(x) dx+\int\limits_{R^n}{\varphi(x) dx}} $$ using the method of stationary phase. Here $(S(t))_{t\in R}$ is once integrated, exponentially bounded group of operators in a Banach space $X$, with generator $A$, which satisfies the condition: For every $x\in X$ there exists $\delta=\delta(x)>0$ such that $\frac{S(t)x}{t^{1/2+\delta}}\to 0$ as $t\to 0$. The function $\varphi (x)$ is infinitely differentiable, defined on $R^n$, with values in $X$, with a compact support supp $\varphi$, the function $f(x)$ is infinitely differentiable, defined on $R^n$, with values in $R$, and $f(x)$ on $\operatorname{supp}\varphi$ has exactly one nondegenerate stationary point $x_0$.

Keywords: Strongly continuous group; once integrated semigroup (group); method of stationary phase

Classification (MSC2000): 47D03; 47D62

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