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

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

and

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

then

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

Particular cases

Convolution kernel

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

See also

Young's inequality for products