Fractional Fefferman--Stein inequalities on infinite rooted k-ary trees
Abstract
We study weighted endpoint estimates for the centered fractional maximal operator on the infinite rooted k-ary tree, T. Our first result is a Fefferman--Stein type estimate w({Mαf>λ}) ≲α,s 1/λ∑x∈ T|f(x)|Mα,sw(x), s>1, with a constant independent of the branching parameter k. Although the proof is inspired by the non-fractional argument of Ombrosi, Rivera-Ríos and Safe in \cite{ORRS}, the fractional normalization introduces a genuinely new scale interaction. At radius r, one must renormalize the function by krα while keeping the dyadic cutoff at kr; this leads to a new summation mechanism involving simultaneously krα and kr(1−α). We then study sharpness and the geometry of admissible weights. We construct counterexamples showing that the estimate cannot in general be reduced to s=1 for any 0<α<1. In the subcritical range 0<α<1/2, we characterize power-type radial weights satisfying the natural fractional A1 condition and prove that the failure persists after any fixed finite number of iterations of Mα. We also prove that Mαw≲ w is necessary for the corresponding weighted weak type estimate but is not sufficient in the subcritical range. Finally, we identify a phase transition: for 1/2≤α<1 no nonzero global weight can satisfy Mαw≲ w.
Related
- Fefferman-Stein Type Distribution Function Inequalities on Product Spaces
- Fefferman--Stein type inequalities via area and maximal functions for Schrödinger operators with applications
- Capacitary domination and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces
- Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted k-Ary Trees
- Revisiting Positive Exponent power weighted Stein-Weiss inequalities in L1
BibTeX
@misc{ibanezfirnkorn2026fractional,
title = {{Fractional Fefferman--Stein inequalities on infinite rooted $k$-ary trees}},
author = {Gonzalo Ibañez-Firnkorn and Jorgelina Recchi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08653},
eprint = {2611.08653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08653},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fractional Fefferman--Stein inequalities on infinite rooted k-ary trees
AU - Gonzalo Ibañez-Firnkorn
AU - Jorgelina Recchi
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08653
UR - https://arxiv.org/abs/2611.08653
AB - We study weighted endpoint estimates for the centered fractional maximal operator on the infinite rooted k-ary tree, T. Our first result is a Fefferman--Stein type estimate w({M_αf>λ}) ≲_α,s 1/λ∑_x∈ T|f(x)|M_α,sw(x), s>1, with a constant independent of the branching parameter k. Although the proof is inspired by the non-fractional argument of Ombrosi, Rivera-Ríos and Safe in \cite{ORRS}, the fractional normalization introduces a genuinely new scale interaction. At radius r, one must renormalize the function by k^rα while keeping the dyadic cutoff at k^r; this leads to a new summation mechanism involving simultaneously k^rα and k^r(1−α). We then study sharpness and the geometry of admissible weights. We construct counterexamples showing that the estimate cannot in general be reduced to s=1 for any 0<α<1. In the subcritical range 0<α<1/2, we characterize power-type radial weights satisfying the natural fractional A_1 condition and prove that the failure persists after any fixed finite number of iterations of M_α. We also prove that M_αw≲ w is necessary for the corresponding weighted weak type estimate but is not sufficient in the subcritical range. Finally, we identify a phase transition: for 1/2≤α<1 no nonzero global weight can satisfy M_αw≲ w.
ER -