# Young's inequality for integral operators

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

## Statement

If ${\displaystyle X}$ and ${\displaystyle Y}$ are measurable spaces, if ${\displaystyle K:X\times Y\to \mathbf {R} }$ and ${\displaystyle {\frac {1}{q}}={\frac {1}{p}}+{\frac {1}{r}}-1}$, if

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

and

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

then [1]

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

## Particular cases

### Convolution kernel

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