Optimality conjecture for the weight classes S for Schrödinger operators

Let $V\in RH_d$. The classes $S_{p,c}^{V}$ are weight classes associated with the reverse Hölder potential $V$, and $R_V$ and $T_V^*$ are the Riesz transform and heat maximal operator, respectively. For a weight $w$, $\|R_V\|_{L^p(w)}<\infty$ and $\|T_V^*\|_{L^p(w)}<\infty$ denote boundedness on $L^p(w)$.

Optimality conjecture. There exist $c_1,c_2>0$ such that

$$S_{p,c_1}^{V}\subset\left\{w:\|R_V\|_{L^p(w)}<\infty\right\}\subset S_{p,c_2}^{V}$$

and

$$S_{p,c_1}^{V}\subset\left\{w:\|T_V^*\|_{L^p(w)}<\infty\right\}\subset S_{p,c_2}^{V}.$$

The conjecture concerns the optimality of the classes $S_{p,c}^{V}$ as characterizations of weighted boundedness. The first inclusion chain is proved for constant potentials, and the second for potentials bounded both above and below; the general case remains open in the supplied text.