Integral operator convergence study in $L^2(\mathbb{R})$

Question

Exercise

I want to study in $\mathcal{H}=L^2(\mathbb{R})$ the convergence of $$A_nf(x)=\log n \int_\mathbb{R} \frac{1}{1+n(x-y)^2}f(y)dy. $$

Solving

a) Pointwise convergence :

For $n \to \infty$ it's easy to see that $A_n\to A=0$ pointwise

b) Strong convergence

I want to show $\|(A_n-A)f\|^2\to 0$ that is $\|A_nf\|^2\to0$. Using Young's inequality , if $f(x)\in L^2(\mathbb{R})$ then $ f(x)^2\in L^1(\mathbb{R})$. Changing variables with $z=\sqrt{n}t$, $$\|A_nf\|_2=\|f*K_n\|_2\le \frac {\log n}{n^{1/2}}\,\|f\|_2\|K\|_1\to0, $$ where $K_n= \frac{1}{1+nt^2}$ and $K= \frac{1}{1+z^2}$.

c) Uniform convergence

Is the Cauchy–Schwarz inequality useful? ...

Answer 1

You have already shown that $$ \|A_nf\|_2\leq\frac {\|K\|_1\,\log n}{\sqrt n}\, \|f\|_2, $$ which says that $$ \|A_n\|\leq\frac {\|K\|_1\,\log n}{\sqrt n}. $$