Sharp local fractional maximal inequality beyond the range q ≤ p'

Determine whether the quantitative bound for the matrix-weighted local fractional maximal operator M_{W,\alpha,r} holds with exponent (p'/q)(1-\alpha/n) on the local matrix Muckenhoupt characteristic [W]_{\mathscr{A}^{\operatorname{loc}_{p,q}(r)}} without the additional assumption q≤p'.

Background

Theorem 3.1 establishes the estimate |M_{W,\alpha,r}|{L^p\to L^q}\lesssim [W]{\mathscr{A}^{\operatorname{loc}_{p,q}(r)}}^{(p'/q)(1-\alpha/n)} under the additional condition q≤p'. The authors prove that this exponent is sharp within that range, but explicitly leave unresolved whether the same estimate remains valid for the full range of indices allowed by the relation 1/q=1/p-\alpha/n.

For \alpha=0, the condition q≤p' is equivalent to p≤2. Thus, the unresolved issue includes the matrix-valued or higher-dimensional local maximal-operator problem for p>2.