Sharpness of the local matrix-weighted fractional integral bound

Determine whether the exponent (p'/q)(1-\alpha/n)+1/q' in the quantitative L^p(W^{p/q})\to L^q(W) bound for the local fractional integral operator I_{\alpha,r} is sharp for matrix weights W\in\mathscr{A}^{\rm loc}_{p,q}(r).

Background

Theorem 3.4 proves a quantitative estimate for the local fractional integral operator I_{\alpha,r} on matrix-weighted spaces, with dependence on the local matrix characteristic raised to (p'/q)(1-\alpha/n)+1/q'. The scalar specialization later achieves the sharp scalar exponent, but this does not settle optimality in the genuinely matrix-weighted setting.

The authors explicitly note that the matrix exponent does not match the known sharp scalar exponent and that, although it specializes formally to the sharp 3/2 exponent in a related matrix Calderón–Zygmund setting when \alpha=0 and p=q=2, its optimality for the local fractional integral estimate remains unresolved.