A question on a simple integral with a singular kernel?

Question

I asked this question on math.stackexchange:

Does this integral converge when $\frac{1}{p}+\frac{1}{q}\ge1$?

No answers or very useful comments there. May be it is more appropraite for mathoverflow.

Fix a small $\delta>0$ and let $p,q>1$. Consider the integral

$$I(p,q):=\int\limits_{1-\delta}^{1+\delta} \int\limits_{y/2}^{2y}\frac{1}{|y-x|^{\frac{1}{p}}|1-x|^{\frac{1}{q}}} \,\mathrm{d}x\,\mathrm{d}y. $$

I am trying to show that $I(p,q)$ diverges if $\frac{1}{p}+\frac{1}{q}\geq 1$. I am not sure this is even the case ? Any hints on how to handle this?

Remark: This seems to be related to the failure of the Hardy-Littlewood-Sobolev inequality (HLS) at the endpoint $p=1$. HLS reads:

If $1<p,q<\infty$, $f\in L^p$ and

$$Tf(x):=\int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{\gamma}}dy$$

Then $$\|Tf\|_q\leq \|f\|_p$$ if and only if $$\frac{1}{p}-\frac{1}{q}=1-\frac{\gamma}{n}.$$

Many thanks.

Answer 1

It converges whenever $p$ and $q$ are both greater than 1. For fixed $x$, the integral against $y$ of $|y-x|^{-1/p}$ is uniformly bounded. Then, $\int_{0}^2 |1-x|^{-1/q}dx$ converges.