Weighted Estimates for Maximal Functions Associated to Skeletons
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Weighted Estimates for Maximal Functions Associated to Skeletons Andrea Olivo1 · Ezequiel Rela1 Received: 14 March 2019 © Mathematica Josephina, Inc. 2019
Abstract We provide quantitative weighted estimates for the L p (w) norm of a maximal operator associated to cube skeletons in Rn . The method of proof differs from the usual in the area of weighted inequalities since there are no covering arguments suitable for the geometry of skeletons. We use instead a combinatorial strategy that allows to obtain, after a linearization and discretization, L p bounds for the maximal operator from an estimate related to intersections between skeletons and k-planes. Keywords Maximal functions · Weights · Skeletons Mathematics Subject Classification Primary: 42B25 · Secondary: 43A85
1 Introduction and Main Results 1.1 Averaging Operators and Packings The purpose of this article is to study weighted estimates for a certain type of maximal operators related to a geometric problem of “packing objects.” The most famous example of this type of problems is the Kakeya needle problem: how small can be a set E ⊂ Rn containing a unit line segment in every possible direction e ∈ Sn−1 ? It is known that those sets can be of null Lebesgue measure, but the question regarding the smallest possible value for the dimension in Rn remains open for n ≥ 3. The best known results on Kakeya with respect to the Hausdorff dimension were obtained as a consequence of L p bounds for an appropriate maximal operator defined by averaging
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Ezequiel Rela [email protected] Andrea Olivo [email protected]
1
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pabellón I, 1428 Buenos Aires, Argentina
123
A. Olivo, E. Rela
over thin tubes pointing in any possible direction. More precisely, the Kakeya maximal function is defined by 1 Kδ ( f )(e) = sup δ (x)| n |T x∈R e
Teδ (x)
| f (x)| dx, e ∈ Sn−1 ,
where Teδ (x) is a 1 × δ-tube (by this we mean a tube of length 1 and cross section of radius δ) centered at x in the direction of e ∈ Sn−1 ⊂ Rn . Due to the existence of zero measure Kakeya sets, this operator can only be bounded on L p with a dependence on δ that blows up when δ → 0. It is precisely this rate of blow up that provides the key ingredient to derive dimension bounds for the Kakeya sets. Other examples involving packing of circles of every possible radii or centered at any point of a prescribed set lead to the study of the corresponding maximal operators. For example, the problem of packing circles is related to the properties of the spherical maximal function Msph f (x) = sup r >0
1 σ (S n−1 (x, r ))
S n−1 (x,r )
| f (y)| dσ,
(1.1)
where σ is the surface measure on the sphere and S n−1 (x, r ) denotes the n − 1 sphere centered on x and with radius r . Results for the boundedness of this operator can be found in [17] for n ≥ 3 and in [1] for the more difficult case of n = 2 (see also [11] and [21] for related problems involving similar maximal operators). In this article, w
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