Fefferman–Stein Inequalities for the Hardy–Littlewood Maximal Function on the Infinite Rooted k-ary Tree

Abstract

In this paper, weighted endpoint estimates for the Hardy–Littlewood maximal function on the infinite rooted $k$-ary tree are provided. Motivated by Naor and Tao [ 23], the following Fefferman–Stein estimate $$\begin{align*}& w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}\: \text{d}x\qquad s>1\end{align*}$$is settled, and moreover, it is shown that it is sharp, in the sense that it does not hold in general if $s=1$. Some examples of nontrivial weights such that the weighted weak type $(1,1)$ estimate holds are provided. A strong Fefferman–Stein-type estimate and as a consequence some vector-valued extensions are obtained. In the appendix, a weighted counterpart of the abstract theorem of Soria and Tradacete [ 38] on infinite trees is established.

Weighted Inequalities for Fractional maximal functions on the infinite rooted $k$-ary tree

“…In recent times there are several substantial works devoted to the study of the discrete analogues of the Hardy-Littlewood maximal function and the fractional Hardy-Littlewood maximal function in various frameworks, for instance we refer [CH12, GRM21, GRM22]. Our main motivation for studying this object originates from the recent works [ORRS21] and [ORR21]. In [ORRS21], Ombrosi, Rivera-Ríos and Safe have proved a sharp analog of Fefferman-Stein inequality for the Hardy-Littlewood maximal operator on the infinite rooted k-ary tree and subsequently in [ORR21] weighted inequalities for the Hardy-Littlewood maximal function were investigated by Ombrosi and Rivera-Ríos.…”

“…Our main motivation for studying this object originates from the recent works [ORRS21] and [ORR21]. In [ORRS21], Ombrosi, Rivera-Ríos and Safe have proved a sharp analog of Fefferman-Stein inequality for the Hardy-Littlewood maximal operator on the infinite rooted k-ary tree and subsequently in [ORR21] weighted inequalities for the Hardy-Littlewood maximal function were investigated by Ombrosi and Rivera-Ríos. We also refer the articles [ST16,ST19] where the authors studied connections between geometrical properties of infinite graphs and the boundedness of the Hardy-Littlewood maximal operator.…”

“…x ∈ T k . This is due to the fact that |S(x, r)| ≃ k r and |B(x,r)| |S(x,r)| ≤ 2 (see [ORRS21]). Keeping this in mind we will only write our results for the spherical fractional maximal function which can be easily transferred to the fractional Hardy-Littlewood maximal function.…”

“…We would also like to note that, in [CMS10], the same result was proved using a different approach. The idea of our proofs is inspired by the works [ORRS21] and [ORR21] where they have brilliantly implemented the techniques developed by Naor and Tao in the weighted setting. We also refer the article [ST16] where the authors extend the result of Naor and Tao to more general infinite graphs.…”

Weighted $L^{p}$ estimates on the infinite rooted $k$-ary tree

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In this paper, building upon ideas of Naor and Tao [13] and continuing the study initiated in [16] by the authors and Safe, sufficient conditions are provided for weighted weak type and strong type (p, p) estimates with p > 1 for the centered maximal function on the infinite rooted k-ary tree to hold. Consequently a wider class of weights for those strong and weak type (p, p) estimates than the one obtained in [16] is provided.

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