Does Mollification Convergence in Lp still hold for Sequences?

Question

Suppose that $f_n \to f \in L^p(\mathbf{R}^N)$ and $\varphi_{\frac{1}{n}}$ is the standard mollifier $\varphi$ normalized to $\frac{1}{n}$. Do we have $$ \lim_{n \to \infty} \| \varphi_{\frac{1}{n}} * f_n - f_n \|_{L^p(\mathbf{R}^N)} = 0? $$ I know that the result is standard if we do not have sequence and I hypothesize the above follows from the standard result. It seems like we should just be adding and subtracting $f$ in the norm and use the middle-man trick and triangle inequality. However, is it true that $\lim_{n \to \infty} \| \varphi_{\frac{1}{n}} * f_n - f \| = 0$?

Answer 1

Instead of only $f$, add and subtract $f+\varphi_{1/n}*f$, so that matters are reduced to showing $$ \| \varphi_{1/n}* (f_n-f)\|_p \to 0. $$ For this, use Minkowski's integral inequality to get $$ \| \varphi_{1/n}* (f_n-f)\|_p \leq \| f_n-f\|_p. $$

Answer 2

Let $g_n=f_n-f$. Then by [1] and the hypothesis $\| \ g_n \|_p \to 0$, we have $$\| \varphi_{\frac{1}{n}} * g_n \|_p \le \|\varphi_{\frac{1}{n}}\|_1 \cdot \| g_n \|_p = \| \ g_n \|_p \longrightarrow 0 $$ as $ n \to \infty$. Thus, by the triangle inequality in $L^p$, \begin{eqnarray} \| \varphi_{\frac{1}{n}} * f_n - f_n \|_p &=&\| \varphi_{\frac{1}{n}} * (f+g_n) - f-g_n \|_p \\ &\le& \| \varphi_{\frac{1}{n}} * f - f \|_p + \| \varphi_{\frac{1}{n}} * g_n \|_p+ \| g_n \|_p \; \longrightarrow 0 \end{eqnarray} as $ n \to \infty$.

[1] Inequality for the p norm of a convolution