Dear Math Overflowers,

I'm looking for references on the parabolic Harnack inequality for **distributional** solutions of the heat equation on the whole space
$$
\partial_t u=\Delta u\quad\text{and}\quad u(0)=u_0\quad\Leftrightarrow\quad u(t,x)=\int\limits_{R^d}\Gamma_t(x-y)u_0(y)dy.
$$
Here $\Gamma_t(z)=\frac{e^{-\frac{|z|^2}{4t}}}{(4\pi t)^{d/2}}$ is the usual heat kernel and my initial data $u_0\geq 0$ is only in $L^1\cap L^p(R^d)$ for some $p>1$.

More precisely, I'm interested in the case when the initial datum $u_0$ is supported on a positive finite measure set $E=supp(u_0)$ and $u_0|_E\geq M>0$ for some constant $M$. I would like to control $u(t,x)\geq (\ldots)$ for points $x\in supp(u_0)$ and small times $t>0$, which really looks like Harnack inequality to me. The thing is that I have no information at all on the initial support (except that it has finite measure so it may not even be bounded) and I only have mere $L^1\cap L^p$ regularity for $u_0$. I strongly doubt that I can get pointwise estimates, but maybe $L^p$ estimates for $|u(t)|_{L^p(E)}\geq (\ldots)$?

Thank you in advance!