Fractional Fefferman--Stein inequalities on infinite rooted $k$-ary trees

arXiv:2611.08653v1 Announce Type: new 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 \[ w\bigl(\{M_\alpha f>\lambda\}\bigr) \lesssim_{\alpha,s} \frac1\lambda\sum_{x\in T}|f(x)|M_{\alpha,s}w(x), \qquad s>1, \] with a constant independent of the branching parameter $k$. Although the proof is inspired by the non-fractional argument of Ombrosi, Rivera-R\'ios and Safe in \cite{ORRS}, the fractional normalization introduces a genuinely new scale interaction. At radius $r$, one must renormalize the function by $k^{r\alpha}$ while keeping the dyadic cutoff at $k^r$; this leads to a new summation mechanism involving simultaneously $k^{r\alpha}$ and $k^{r(1-\alpha)}$. 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 $0<\alpha<1$. In the subcritical range $0<\alpha<1/2$, 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 $M_\alpha$. We also prove that $M_\alpha w\lesssim w$ is necessary for the corresponding weighted weak type estimate but is not sufficient in the subcritical range. Finally, we identify a phase transition: for $1/2\le\alpha<1$ no nonzero global weight can satisfy $M_\alpha w\lesssim w$.

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