Endpoint equality in the sparse form exponent

Determine whether the sparse form conclusion of Theorem 1 can be strengthened by taking the exponent q equal to q_0, thereby obtaining a (1,q_0')-sparse form bound under the theorem’s hypotheses.

Background

The main theorem establishes a (1,q')-sparse form bound for every q in the open interval (1,q_0), where q_0 is determined by the weighted Lr assumptions on the operator. Equality q=q_0 would provide the endpoint sparse bound excluded by the theorem as stated.

The paper notes that no counterexample is currently known. Resolving this question would also improve the sparse interpolation theorem presented later in the paper and could yield sharper weighted estimates.

References

One can of course speculate whether or not Theorem~\ref{main} can be further improved. In particular, one could investigate whether or not $q$ can be taken equal to $q_0$ in ineq:sparseconclusion. To the best of the author's knowledge there is currently no counter-example to this in the literature.

ineq:sparseconclusion:

∣⟨T(f),g⟩∣≤C∑Q∈S∣Q∣⟨f⟩1,Q⟨g⟩q′,Q.\left|\big\langle T(f),g\big\rangle\right| \leq C \sum_{Q\in\mathcal{S}} |Q|\langle f \rangle_{1,Q}\langle g \rangle_{q',Q}.

— Rubio de Francia's Extrapolation Theorem and Sparse Bounds  (2608.22988 - Rule, 24 Aug 2026) in Remark immediately following the proof of Theorem 1, Section 1