I'm struggeling to prove the following statement: Show that there exists a constant $C>0$ such that for all compactly supported, continuous and integrable functions $f\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$ there holds \begin{align*} \int_{B_1(0)}\int_{B_1(0)}F(x,y)\,dx\,dy\leq C\int_0^1\int_{\partial B_r(0)} \int_{\partial B_r(0)}F(x,y)|x-y|\,dH^{n-1}(x) \, dH^{n-1}(y) \, dr \end{align*} If I write out the left side into two double integrals, then I have to integrals from zero to one and two boundary integrals, and I do not know how to get fruitfully get rid of one "radius" integral.

Any comments or ideas would be deeply appreciated!

Best, Jim


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