Young's inequality for integral operators

In mathematical analysis, the Young's inequality for integral operators, is a bound on the $$L^p\to L^q$$ operator norm of an integral operator in terms of $$L^r$$ norms of the kernel itself.

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

\int_{Y} |K (x, y)|^r \,\mathrm{d} y \le C^r $$ for all $$ x\in X $$ and

\int_{X} |K (x, y)|^r \,\mathrm{d} x \le C^r $$  for all $$ y\in Y $$ then

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

Convolution kernel
If $$X = Y = \mathbb{R}^d$$ and $$K (x, y) = h (x - y) $$, then the inequality becomes Young's convolution inequality.