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 \[ w\bigl(\{M_αf>λ\}\bigr) \lesssim_{α,s} \frac1λ\sum_{x\in T}|f(x)|M_{α,s}w(x), \qquad s>1, \] with a constant independent of the branching parameter $k$. Although the proof is inspired by the non-fractional argument of Ombrosi, Rivera-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 $M_αw\lesssim w$ is necessary for the corresponding weighted weak type estimate but is not sufficient in the subcritical range. Finally, we identify a phase transition: for $1/2\leα<1$ no nonzero global weight can satisfy $M_αw\lesssim w$.