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 $q,p,r\ge 1$ are such that $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}-1$. If

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

and

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

then

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

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