Young's Convolution Inequality

Suppose \(p,q,r\in [1,\infty ]\) with

\[ \frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}. \]

Then for \(f\in L^p(\mathbb{R}^d)\) and \(g\in L^q(\mathbb{R}^d)\) the Young’s inequality hold

\[ \lVert f\ast g\rVert_{L^r(\mathbb{R}^d)}\le \lVert f\rVert_{L^p(\mathbb{R}^d)}\lVert g\rVert_{L^q(\mathbb{R}^d)}. \]