Let $p\in[1,\infty]$. For $f\in L^p(0,\infty)$ we define $Tf:x\mapsto \int_0^\infty K(x,y)f(y)\,dy$ where $K$ is homogenous of degree $-1$, i.e. $K(\lambda x,\lambda y) = \lambda^{-1} K(x,y)$ for $\lambda>0$. Suppose that $$A_K=\int_0^\infty |K(1,y)|y^{-1/p}\,dy<\infty.$$
Then $\|Tf\|_p\leq A_K\|f\|_p$.
My attempts included applying the generalised Minkowski-inequality and dualizing $L^p=(L^q)^*$ but the farthest I've gotten is showing that $$\|Tf\|_p\leq \int_0^\infty |f(y)|\left( \int_0^\infty |K(x,y)|^pdx\right)^{1/p}dy$$ but it doesn't exactly seem promising.
I'm extremely thankful for any help!