(Please, what is going on?) The following claim is from a paper which was apparently reviewed by László Erdös, Zhongwei Shen, and Bernard Heffler. Someone tell me it's true. Surely it's true. The entire paper depends on it being true.

I am looking at a 2000 article by Kazuhiro Kurata in the J. London Math. Soc.: An Estimate on the Heat Kernel of Magnetic Schrödinger Operators and Uniformly Elliptic Operators with Nonnegative Potentials. The claim is Lemma 1 on page 893. (In the preprint, available online here, the claim appears more casually, inside a proof on the top of page 12.)

Kurata is studying the heat kernels of various Schrödinger operators, the simplest example being $H = -\Delta + V(x)$. His key condition is that $V(x)$ belongs to a reverse Hölder class $RH_{q}$ with $q > n/2$.

Now suppose $u(x,t)$ is a weak solution of $(\partial_{t} + H)u = 0$ in some cylinder $$Q_{2r}(x_{0}, t_{0}) = B(x_{0}, 2r) \times (t_{0}-(2r)^{2}, t_{0})$$

According to Kurata, under just these hypotheses, there is a ``standard subsolution estimate''

$$\sup_{(x,t) \in Q_{r/2}(x_{0}, t_{0})} |u| \leq \biggl(\frac{C}{r^{n+2}} \iint_{Q_{2r/3}(x_{0},t_{0})}|u|^2\,dx\,dt\biggr)^{1/2}$$

**with $C$ independent of $V$, $t_{0}$, and $r$.** He cites Aronson and Serrin's foundational '67 paper ``Local behavior of solutions of quasilinear parabolic equations'', but gives no specifics.

In every local boundedness estimate I've seen for parabolic equations (including, incidentally, Theorem 2 in Aronson and Serrin '67), the constants seem to depend on $V$, $t_{0}$, and $r$...!?! Is there just something magical about the reverse Hölder hypothesis?

Life, I guess.