Geometric properties of infinite graphs and the Hardy–Littlewood maximal operator

Soria, Javier; Tradacete, Pedro

doi:10.1007/s11854-019-0019-5

Cited by 23 publications

(21 citation statements)

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“…At this point we would like to mention works by Soria and Tradacete [31,32] in which they study the connection between properties of the maximal function and properties of the underlying graphs. Furthermore [32,Theorem 4.1] is an abstract version of Theorem A.…”

Section: Intoduction and Main Resultsmentioning

confidence: 99%

“…In this appendix we provide a weighted version of [32,Theorem 4.1]. Analogously to the case of the infinite rooted k-ary tree the spherical maximal function on any tree T can be defined as follows: Again, as we did for the infinite rooted k-ary tree, given any infinite tree T we can define the average operator over the tree T as…”

Section: Appendix a A Weighted Version Of Soria-tradacete Results For Infinite Treesmentioning

confidence: 99%

“…Let α ∈ (0, 1) and s ∈ (1, ∞). We define E w T (s, r, α) = sup E,F ⊂G |E|,|F |<∞ 1 w(F ) α M • s w(E) 1−α x∈E w (F ∩ S(x, r)) |S(x, r)| and S T (r) = sup x∈T |S(x, r)|.With these quantities at our disposal we are ready to settle our weighted version of[32, Theorem 4.1]. For every weight w on a tree T we have thatw ({x ∈ T : M • f (x) > λ}) Γ T,w,r,α,s 1 λ T |f(x)|M • s w(x)dx where Γ T,w,r,α,s = c α sup n∈N    ∞ S T (r)≥2 n−1 E w T (s, r, α) 1 1−α S T (r) 1 and c α → +∞ when α → 0.…”

mentioning

confidence: 99%

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

Ombrosi

Rivera-Rı́os

Safe

2020

International Mathematics Research Notices

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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.

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Section: Intoduction and Main Resultsmentioning

confidence: 99%

Section: Appendix a A Weighted Version Of Soria-tradacete Results For Infinite Treesmentioning

confidence: 99%

mentioning

confidence: 99%

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

Ombrosi

Rivera-Rı́os

Safe

2020

International Mathematics Research Notices

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“…For instance, it is easy to check that for the metric measure spaces considered here, the centered maximal operator acting on just one Dirac delta is weak (1,1) bounded, with bound C µ (this is so even if the Dirac delta is placed at a point outside the support of µ). The special case of this result for graphs appears in [SoTr,Proposition 2.10].…”

Section: Local Comparability Vs Doublingmentioning

confidence: 96%

Local Comparability of Measures, Averaging and Maximal Averaging Operators

Aldaz

2017

Potential Anal

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We explore the consequences for the boundedness properties of averaging and maximal averaging operators, of the following local comparabiliity condition for measures: Intersecting balls of the same radius have comparable sizes. Since in geometrically doubling spaces this property yields the same results as doubling, we study under which circumstances it is equivalent to the latter condition, and when it is more general. We also study the concrete case of the standard gaussian measure, where this property fails, but nevertheles averaging operators are uniformly bounded, with respect to the radius, in L 1 . However, such bounds grow exponentially fast with the dimension, very much unlike the case of Lebesgue measure.

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“…Moreover, there are (infinite) graphs G where no doubling measure exists (i.e., C G = ∞), even though G is always a complete metric space. For example, it is easy to see that this is the case for the k-homogeneous tree T k , k ≥ 3, since it is not doubling in the metric sense [15].…”

Section: Examplesmentioning

confidence: 99%