In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young.

In real analysis, the following result is called Young's convolution inequality:

Suppose ${\displaystyle f}$ is in the Lebesgue space ${\displaystyle L^{p}(\mathbb {R} ^{d})}$ and ${\displaystyle g}$ is in ${\displaystyle L^{q}(\mathbb {R} ^{d})}$ and

$${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1}$$

with ${\displaystyle 1\leq p,q,r\leq \infty .}$ Then

$${\displaystyle \|f*g\|_{r}\leq \|f\|_{p}\|g\|_{q}.}$$

Here the star denotes convolution, ${\displaystyle L^{p}}$ is Lebesgue space, and

$${\displaystyle \|f\|_{p}={\Bigl (}\int _{\mathbb {R} ^{d}}|f(x)|^{p}\,dx{\Bigr )}^{1/p}}$$