Convolution

(Properties of convolution from Fubini--Tonelli)

The integrability of \(f*g\), associativity, and Young’s inequality are proved by applying Tonelli's or Fubini's theorem to the function \((x,y)\mapsto f(x-y)g(y)\) on \(\RR^n\times\RR^n\):

• for \(f,g\in L^1\), Tonelli’s theorem applied to \(\abs{f(x-y)g(y)}\) gives \(\int\int\abs{f(x-y)g(y)}\dy\dx = \norm f_1\norm g_1\), so \(f*g(x)\) is defined for almost every \(x\) and \(\norm{f*g}_1 \leq \norm f_1\norm g_1\);

• commutativity \(f*g=g*f\) follows from the change of variables \(y \mapsto x-y\), and associativity \((f*g)*h=f*(g*h)\) for \(f,g,h\in L^1\) from that change of variables and Fubini’s theorem;

• Young’s inequality \(\norm{f*g}_r \leq \norm f_p\norm g_q\) with \(\frac1r = \frac1p + \frac1q - 1\) follows from Hölder’s inequality applied inside Tonelli’s theorem.

(Smoothing)

If \(f\in L^1(\RR^n)\) and \(g\in C^k(\RR^n)\) with \(\partial^\alpha g\) bounded for \(\abs\alpha\leq k\), then \(f*g\in C^k\) and \(\partial^\alpha(f*g) = f*\partial^\alpha g\), by differentiating under the integral with the dominating function \(\abs f\,\norm{\partial^\alpha g}_\infty\). With \(\phi\in C_c^\infty\), \(\int\phi=1\), and \(\phi_t(x)\coloneqq t^{-n}\phi(x/t)\), the functions \(f*\phi_t\) are smooth and converge to \(f\) in \(L^p\) as \(t\to0\) for \(f\in L^p\), \(1\leq p<\infty\); this proves density of \(C^\infty\) functions in \(L^p\).