Convolution with an $L^1$ function acts boundedly on every $L^p$.Suggest a slogan
Let \(1\leq p\leq\infty\), \(f\in L^p(\RR^n)\), and \(g\in L^1(\RR^n)\). Then the convolution \(f*g\) is defined almost everywhere, \(f*g\in L^p(\RR^n)\), and \[ \norm{f \ast g}_p \leq \norm{f}_p \norm{g}_1 . \] This is the case \(q=1\), \(r=p\) of Young’s convolution inequality PR-2C3SZ.