Convergence of $\theta_n * f$ to $f$ in $L^p$ for $p<1$ for $\theta_n$ mollifiers?

Question

Consider a compactly supported smooth function $\theta \geq 0$ defined over $\mathbb{R}$ satisfying $\int_{\mathbb{R}} \theta (t) \mathrm{d} t = 1$ and set, for $n\geq 1$, $\theta_n(t) = n \theta( n t)$. Let $f$ be a bounded measurable function on $\mathbb{R}$.

If $p \geq 1$, then $\theta_n * f$ converges to $f$ in $L^p[0,1]$. One can show this using the Young inequality.

Is the result still valid when $p<1$ and how to prove it?

Thanks.

Answer 1

On a set $\Omega$ of finite measure, the integral $\int_\Omega |g|^p$ is controlled by $\int_\Omega |g|^q$ whenever $p<q$, the reason being Hölder's inequality: $$ \int_\Omega |g|^p = \int_\Omega 1^{1-p/q}(|g|^q)^{p/q} \le \left(\int_\Omega 1\right)^{1-p/q} \left(\int_\Omega |g|^q\right)^{p/q} $$ Hence, if $\int |f-\theta_n*f|$ converges to $0$ (which it does for any locally integrable $f$), so does $\int |f-\theta_n*f|^p$ for $0<p<1$.