Let \((X,\mathcal M,\mu)\) and \((Y,\mathcal N,\nu)\) be sigma-finite measure spaces. Let \(K:X\times Y\to\mathbb C\) be measurable and suppose \[ \int_Y |K(x,y)|\,d\nu(y)\le A \quad\text{for all }x, \] and \[ \int_X |K(x,y)|\,d\mu(x)\le A \quad\text{for all }y. \] Define \[ Tf(y)=\int_X f(x)K(x,y)\,d\mu(x). \] Prove that for every \(1\le p\le\infty\), \[ \|Tf\|_{L^p(\nu)}\le A\|f\|_{L^p(\mu)}. \]
Solution
1. The case \(p=\infty\).
For almost every \(y\), \[ |Tf(y)|\le \|f\|_\infty\int_X|K(x,y)|\,d\mu(x)\le A\|f\|_\infty. \] Thus \(\|Tf\|_\infty\le A\|f\|_\infty\).
2. The case \(1\le p<\infty\).
Fix \(y\). Hölder’s inequality with respect to the finite measure \(|K(x,y)|\,d\mu(x)\) gives \[ |Tf(y)|^p \le\left(\int_X |f(x)|^p|K(x,y)|\,d\mu(x)\right) \left(\int_X |K(x,y)|\,d\mu(x)\right)^{p-1}. \] By the kernel bound, \[ |Tf(y)|^p\le A^{p-1}\int_X |f(x)|^p|K(x,y)|\,d\mu(x). \] Integrating in \(y\) and using Tonelli, \[ \begin{aligned} \|Tf\|_p^p &\le A^{p-1}\int_Y\int_X |f(x)|^p|K(x,y)|\,d\mu(x)\,d\nu(y)\\ &=A^{p-1}\int_X |f(x)|^p\left(\int_Y|K(x,y)|\,d\nu(y)\right)d\mu(x)\\ &\le A^p\|f\|_p^p. \end{aligned} \] Taking \(p\)th roots gives \[ \|Tf\|_p\le A\|f\|_p. \]