This problem is from exercise 30 on page 196 of Folland's "Real Analysis". Let $K$ be a non-negative measurable function on $[0,\infty)$ and let
$\phi(s) = \displaystyle\int_0 ^\infty K(x)x^{s-1} dx$.
I am trying to prove that if $f$ and $g$ are non-negative measurable functions and $p$ and $q$ are conjugate exponents, then
$\displaystyle\int_0 ^\infty \int_0 ^\infty K(xy)f(x)g(y) dxdy \leq \phi(1/p) \left(\int_0 ^\infty x^{p-2} f(x)^p dx\right)^{1/p} \left(\int_0 ^\infty g(x)^q dx\right)^{1/q}$.
So far, I have been attempting various combinations of Hölder's inequality and the theorem given in the text for a function $K$ on $\mathbb R^2$ which is homogenous of degree $-1$, but have not met with much success. Any suggestions?