The Method of Stationary Phase for Once Integrated Group

Ramiz Vugdalić   ; Fikret Vajzović

Publications de l'Institut Mathématique, _N_S_79 (2006) no. 93, p. 73

Résumé

We obtain a formula of decomposition for $ \Phi(A)=A\intłimits_{R^n}{S(f(x))\varphi(x) dx+\intłimits_{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 $x_0$.

Détail

Classification : 47D03 47D62
Keywords: Strongly continuous group, once integrated semigroup (group), method of stationary phase