Determine whether the local real-valued vanishing-order exponent can be reduced to one-half

Determine whether the exponent $2/3$ in the quantitative vanishing-order bound for real-valued solutions $w$ and real-valued potentials $Q$ in the normalized local Schrödinger problem can be replaced by $1/2$.

Background

For solutions of $\Delta w=Qw$ in $B_{10}$ satisfying $\|Q\|_{L^\infty(B_{10})}\le A$, $\|w\|_{L^\infty(B_6)}\le1$, and $\|w\|_{L^\infty(B_1)}\ge1$, the paper recalls the bound $\operatorname{ord}_p w\le C(1+A^{2/3})$ for $p\in B_1$. The cited question concerns whether the exponent can be improved to the square-root exponent for real-valued $w$ and $Q$.

The paper’s construction establishes sharpness of the $2/3$ exponent for a fixed-ball Dirichlet problem, but explicitly distinguishes that setting from the normalized local problem, so it does not resolve this local question.