Sharpness of the local Calderón–Zygmund bound

Determine whether the exponent 1+1/(p-1)-1/p in the quantitative L^p(W) bound for a (\delta,t)-Calderón–Zygmund operator with exponential decay is sharp for local matrix Muckenhoupt weights W\in\mathscr{A}^{\operatorname{loc}_{p}(r)}.

Background

Theorem 3.10 establishes boundedness of Calderón–Zygmund operators with exponential decay on L^p(W), with quantitative dependence [W]{\mathscr{A}^{\operatorname{loc}{p}(r)}}^{1+1/(p-1)-1/p}. The exponent agrees with the corresponding global matrix-weighted Calderón–Zygmund exponent and equals 3/2 when p=2.

Despite this agreement and the known sharpness of the global 3/2 exponent for the Hilbert transform, the authors do not establish sharpness for the local exponential-decay estimate itself.