Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg\int_{\mathbb{R}^3}dx\,u(x)^2(1+x^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$) I have to calculate this limit $$\lim_{xy\to 0}\int_{\mathbb{R}^3}\frac{e^{i\lambdaxx'}}{xx'}f(x')dx'$$ with $\lambda>0$ and $y\in\mathbb{R}^3$. I've thought to use the theorem of dominated convergence, so I note that the integrand converges to $$\frac{e^{i\lambdayx'}}{yx'}f(x')$$ but I don't succed in finding a majoration of it with an integrable function not depending on $xy$. Some ideas?

I think the limit of the integrals does equal the integral of the limit. First of all, outside of a small ball about $y$ we have dominated convergence. So the issue is whether $$\lim_{xy \to 0} \int_{ball(y,\epsilon)} \frac{e^{i\lambdaxx'}}{xx'} f(x')dx'$$ equals the same integral with $y$ in place of $x$. Thus we can ignore the exponential factor  when $x$ gets close to $y$ it's essentially constant on this ball  and the weight $(1 + x^2)^s$ which governs behavior at infinity but is irrelevant near $y$. Fix $\epsilon > 0$ and let $h_x(x') = \frac{1}{xx'}\cdot \chi_{ball(y,\epsilon)}$. I claim that $h_x \to h_y$ weakly in $L^2$ as $x \to y$. This will imply that $\int h_xf \to \int h_yf$, which is all you need for the reasons I just explained. Well, the functions $h_x$ are uniformly bounded in $L^2$, so weak convergence will follow from weak convergence when integrated against a dense subset of $L^2$. But $\int h_xg \to \int h_yg$ for any $L^2$ function $g$ which is $0$ on a neighborhood of $y$, by dominated convergence, and such functions $g$ are dense, so we are done. (In fact weak convergence plus the fact that $\h_x\ \to \h_y\$ implies that $h_x \to h_y$ in norm, which presumably you could show by direct calculation.) 


I had an idea. I know that the function $$F(x)=\int_{\mathbb{R}^3}\frac{e^{i\lambdaxx'}}{xx'}f(x')dx'\in H^{2,s}(\mathbb{R}^3)$$ when $f\in L^{2,s}(\mathbb{R}^3) $for $s>\frac{1}{2}$ (I've proved this fact); moreover the space $H^{2,s}(\mathbb{R}^3)$ is conteined in $C(\mathbb{R}^3)$ and so the considered limit exists. 

