Young's inequality for integral operators

Assume that ${\displaystyle X}$ and ${\displaystyle Y}$ are measurable spaces, ${\displaystyle K:X\times Y\to \mathbb {R} }$ is measurable and ${\displaystyle q,p,r\geq 1}$ are such that ${\displaystyle {\frac {1}{q}}={\frac {1}{p}}+{\frac {1}{r}}-1}$. If

${\displaystyle \int _{Y}|K(x,y)|^{r}\,\mathrm {d} y\leq C^{r}}$ for all ${\displaystyle x\in X}$

and

${\displaystyle \int _{X}|K(x,y)|^{r}\,\mathrm {d} x\leq C^{r}}$ for all ${\displaystyle y\in Y}$

then ^[1]

${\displaystyle \int _{X}\left|\int _{Y}K(x,y)f(y)\,\mathrm {d} y\right|^{q}\,\mathrm {d} x\leq C^{q}\left(\int _{Y}|f(y)|^{p}\,\mathrm {d} y\right)^{\frac {q}{p}}.}$