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Let $\phi(x,\theta)$ be an infinitely differentiable function in $X^2 = X \times X$, where $X = \mathop{\mathsf{int}}\mathbb{R}^n_{+}$, and let$\phi(\lambda x, \theta) = \phi(x,\lambda \theta) = \lambda \phi(x,\theta)$ for any $\lambda > 0$ (positive homogenity). Define $\Phi(x,y,\theta)=\phi(x,\theta)-\phi(y,\theta)$. Let $A(x,y,\theta)$ be an infinetely differentiable function in $X^3 = X\times X \times X$ such that $A(x,y,\theta)=0$ for any $\theta$ with $|\theta|<\varepsilon$ and $A(x,y,\theta)$ and any its partial derivative of any order exponentially decreases as a function of $\theta$ uniformly with respect to $(x,y)$ on any compact set $K\subset X^2$. I define an operator $F$ which acts on functions from $C_{0}(X)$ by the rule $$ F[u](y) = \int\limits_{X} \int\limits_{X} e^{i \Phi(x,y,\theta)} A(x,y,\theta) u(x) \, dx\,d\theta $$ I have two questions.

Question 1 Is it true that $F[u](y) \in C^{1}(X)$, but $F[u](y) \notin C^{2}(X)$ in general?

$\square$ Let $K$ be a support of $u(x)$, then there exist $C(K)$ and $\alpha(K)>0$ such that $$ |A(x,y,\theta)| \leqslant C(K) e^{-\alpha(K)\theta} $$ for any $x \in K$, $y \in K$. This proves the absolute and uniform convergence of integral $F[u](y)$ on any compact set $K \subset X$ and hence we obtain continuity of $F[u](y)$. Next, the same arguments can be used to show that we can differentiate $F[u](y)$ under the integral sign and that derivative is continuous. If I'm not mistaken we can't continue this procedure in general because of decreasing degrees of homogeneity of derivatives of $\Phi$. $\blacksquare$

Now lets introduce a set $$ C_{\Phi} = \left\{ (x,y,\theta) \in X^3 \mid \Phi_{\theta}(x,y,\theta)=0 \right\}. $$

Question 2 How to show that if $C_{\Phi} = \left\{(x,x,\theta) \mid x \in X, \theta \in X \right\}$ then operator $F[u](x)$ can be extended to the operator from $L^2_{0}(X)$ to $L^{2}_{loc}(X)$ such that $F = I + K$, $I$ is the identity operator, $K$ is compact?

I have absolutely no ideas concerning second question, any help is very appreciated. Actually I don't feel deeply the role of condition on set $C_{\Phi}$. Please help me with it.

Update (15.02.13) Let's show that $F \in \mathcal{L}(L^2_{0},L^{2}_{loc})$.

$\square$ It sufficient to show that for any $g \in C^{\infty}_{0}(X)$ there exists $C>0$ and a compact $K$ in $X$ such that for any $u \in L^{2}(K)$ we have $\|gF[u]\| \leqslant C \|u\|$. Let $h(x) \in C^{\infty}_{0}(X)$, $h(x)=1$ in some heighborhood of $X$. Then it is sufficient to show that $\|gF[hu]\| \leqslant C \|u\|$, on the left we have an integral operator with smooth kernel $$ e^{i\Phi(x,y,\theta)}A(x,y,\theta)g(x)h(y) $$ with compact support with respect to $(x,y)$. So without loss of generality we can consider the case when $A(x,y,\theta)$ has a compact support $K_1 \times K_2$ with respect to $(x,y)$. By definition of $A(x,y,\theta)$ there exist such $C_1>0$ and $\alpha>0$ that $$ |A(x,y,\theta)| \leqslant C_1 e^{-\alpha |\theta|} $$ for any $x,y \in X$. Then we have an estimate $$ |F[u](x)| \leqslant C_1 \chi_{K_1}(x) \int\limits_{X} e^{-\alpha |\theta|} \, d\theta \, \int\limits_{K_2} u(y) \, dy \leqslant C_2 |K_2| \chi_{K_1}(x) \|u\| $$ From this we get the required inequality. $\blacksquare$

Now the only question is why $F$ is a sum of the identity operator and of a compact operator.

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