Show that if \(\phi\) is an approximate identity, then \[\norm{f\ast \phi_t - f}_1 \converges{t\to 0}\to 0.\]
- Show that if additionally \(\abs{\phi(x)} \leq c(1 + \abs{x})^{-n-\eps}\) for some \(c,\eps>0\), then this converges is almost everywhere.
Show that is \(f\) is bounded and uniformly continuous and \(\phi_t\) is an approximation to the identity, then \(f\ast \phi_t\) uniformly converges to \(f\).
Solution
Let \(\phi \in L^1(\RR^n)\) with \(\int\phi = 1\) and \(\phi_t(x) = t^{-n}\phi(x/t)\). Then \[ f\ast\phi_t(x) - f(x) = \int\phi(y)\big(f(x - ty) - f(x)\big)\,dy \] by the substitution \(u = ty\) in \(\int f(x-u)\phi_t(u)\,du\) and \(\int\phi = 1\).
1. For \(f \in L^1\), \(\|f \ast \phi_t - f\|_1 \to 0\) as \(t \to 0\).
By Tonelli’s theorem, \(\|f\ast\phi_t - f\|_1 \le \int|\phi(y)|\,\omega(ty)\,dy\) with \(\omega(h) = \int|f(x-h) - f(x)|\,dx\). Continuity of translation in \(L^1\) gives \(\omega(ty) \to 0\) for each \(y\), and \(|\phi(y)|\,\omega(ty) \le 2\|f\|_1|\phi(y)|\), so dominated convergence applies. See E-QCYEM.
2. If also \(|\phi(x)| \le c(1 + |x|)^{-n-\eps}\), then \(f \ast \phi_t \to f\) a.e. for \(f \in L^1\).
2.1. There is \(A\) with \(\sup_{t>0}|f \ast \phi_t(x)| \le A\,Mf(x)\) for every \(x\), where \(Mf\) is the Hardy–Littlewood maximal function.
\(\Psi(x) = c(1+|x|)^{-n-\eps}\) is a radial, radially decreasing, integrable majorant of \(|\phi|\). For such \(\Psi\), \(\sup_t (|f| \ast \Psi_t)(x) \le \|\Psi\|_1\,Mf(x)\), by writing \(\Psi\) as an increasing limit of positive combinations of normalized indicators of balls centered at \(0\). Take \(A = \|\Psi\|_1\).
2.2. Q.E.D.
2.1. For \(\alpha > 0\), the weak type \((1,1)\) bound \(m\theset{Mh > \beta} \le 3^n\|h\|_1/\beta\) and Chebyshev’s inequality give \(m\theset{\Omega f > 2\alpha} \le (3^nA + 1)\|f - g\|_1/\alpha\). Since \(C_c\) is dense in \(L^1\), \(m\theset{\Omega f > 2\alpha} = 0\) for every \(\alpha > 0\). See E-KUOXT.
3. If \(f\) is bounded and uniformly continuous, then \(f \ast \phi_t \to f\) uniformly.
Given \(\eps > 0\), choose \(M\) with \(2\|f\|_\infty\int_{|y|>M}|\phi| < \eps/2\), and by uniform continuity choose \(t_0\) with \(|f(x - u) - f(x)| < \eps/(2\|\phi\|_1)\) for all \(x\) and \(|u| \le t_0M\). For \(t < t_0\), splitting \(\int|\phi(y)|\,|f(x-ty) - f(x)|\,dy\) at \(|y| = M\) bounds it by \(\eps/2 + \eps/2\) for every \(x\).