Mixed-norm BrascampLieb inequalities arbitrarily close to the endpoint q=1

Establish that for every real number \alpha>1 there exist positive integers and exponents m, p_1, \ldots, p_m, q, r and rational one-dimensional projections B_1, \ldots, B_m of \mathbb{R}^2, with 1/q+1/r=1 and q<\alpha, for which the mixed-norm BrascampLieb inequality holds.

Background

The paper proves that if the mixed-norm BrascampLieb inequality holds for a suitable rational projection configuration with q in (1,2), then Kakeya sets in every dimension satisfy a corresponding Hausdorff-dimension lower bound. It then proposes the conjecture that such inequalities exist with q<\alpha for every \alpha>1. The authors explain that this would yield q arbitrarily close to 1 and consequently imply the full Hausdorff Kakeya conjecture.

References

It seems reasonable to the author to make the following conjecture, which, by Theorem \ref{BLmiximplieskakeya}, will imply Conjecture \ref{KakeyaH}.

For arbitrary $\alpha > 1$, BLmix holds for some choices of $m, p_j, q, r$ and rational projections $B_j$ with assumptionofqandr satisfied and $q < \alpha$.

BLmix:

∥∏j=1mfj(Bj(x,y))pj∥LxqLyr≤C∏j=1m∥fj∥L1pj\|\prod_{j=1}^m f_j (B_j (x, y))^{p_j}\|_{L_x^q L_y^r} \leq C \prod_{j=1}^m \|f_j\|_{L^1}^{p_j}

assumptionofqandr:

1q+1r=1.\frac{1}{q}+ \frac{1}{r} = 1.

— Mixed-norm Brasamp-Lieb inequalities  (2608.17952 - Zhang, 18 Aug 2026) in Section 5.2, subsection “The connection to (BLmix),” Conjecture 5.4