Convolution with Gaussian, without dstributioni theory, part 3

Question

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following:

Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ u(t,x)=\int_{\mathbb{R}^n}\Gamma(t,x-y)f(y)dy. $$ If $p=\infty$, then $u(t,\cdot)\to f$ a.e. as $t\to 0$.

I've shown that $||u(t,\cdot)||_p\leq ||f||_p$ for $1\leq p\le\infty$ but haven't gotten anywhere from here. Could someone please give some hints or suggest a reference (these seem to be very well-known and widely applicable results).

(This is the first, and this is the second part of my question.)

Answer 1

Well, it seems that Folland's Real Analysis contains a Theorem that applies in this case: My edition is the Second One, so you might look at Thm. 8.15 for an idea for the proof for an even more general setting (the same problem is true if you only require that $p\in [1,+\infty]$).