This is a problem in the S.-T. Yau College Student Mathematics Contests in 2013.

Suppose $H=L^2(B)$, $B$ is the unit ball in $\mathbb{R}^d$. Let $K(x,y)$ be a measurable function on $B\times B$ that satisfies $$|K(x,y)|\le A|x-y|^{-d+\alpha}$$ for some $\alpha>0$, whenever $x,y\in B$. Define $$Tf(x)=\int_B K(x,y)f(y){\rm d}y$$ (a) Prove that $T$ is bounded operator on $H$.

(b) Prove that $T$ is compact.

I think this is just a simple manipulation of the Cauchy-Schwarz inequality and Arzela-Ascoli theorem. However, when I employ the Cauchy-Schwarz inequality, I have to estimate $\int_B |K(x,y)|^2{\rm d}y$. In the case $d=1$, if we just set $|K(x,y)|=A|x-y|^{-1+\alpha}$, this would be invalid since $|x|^{-1+\alpha}$ is not square integrable on $[-1,1]$ given $0<\alpha<\frac{1}{2}$.

  • $\begingroup$ Please set a meaningful title. $\endgroup$
    – user65203
    Apr 5, 2019 at 8:51


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