Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided $\frac 1p + \frac 1q = 1 + \frac 1r$ and $1\le p,q,r\le\infty$. Obvious implications is that
- $L^1$ is a Banach algebra
- $L^\infty$ is an $L^1$-module with respect to convolution
Do you have any deeper applications/examples?