Systems of dyadic cubes in a doubling metric space

Hytönen, Tuomas; Kairema, Anna

doi:10.4064/cm126-1-1

Cited by 253 publications

(262 citation statements)

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“…The authors would like to thank the referees for their careful review as well as many valuable comments and suggestions, and for bringing our attention to the references [3,32,37,38].…”

Section: Acknowledgmentsmentioning

confidence: 99%

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On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

Dong

Kim

2018

Trans. Amer. Math. Soc.

120

155

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Abstract. We prove generalized Fefferman-Stein type theorems on sharp functions with Ap weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixednorm weighted Lp-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains. IntroductionThe objective of this paper is two-fold. The first is to present a few generalized versions of the Fefferman-Stein theorem on sharp functions. One of our main theorems in this direction proveswhere (X , µ) is a space of homogeneous type, w is a Muckenhoupt weight, f # dy is the sharp function of f using a dyadic filtration of partitions of X , and L p,q (X , w dµ) is a mixed norm. A space X of homogeneous type is equipped with a quasi-distance and a doubling measure µ. A brief description of spaces of homogeneous type is given in Section 2. For more discussions, see [45,44,11]. If X is the product of two spaces (X 1 , µ 1 ) and (X 2 , µ 2 ) of homogeneous type, w = w 1 (x ′ )w 2 (x ′′ ), and µ(x) = µ 1 (x ′ )µ 2 (x ′′ ), where x ′ ∈ X 1 and x ′′ ∈ X 2 , then the L p,q (X , w dµ) norm is defined asSee (2.13) for a precise definition of the mixed norm. If X is the Euclidean space R d , µ is the Lebesgue measure on R d , w ≡ 1, and p = q, the inequality (1.1) is the celebrated Fefferman-Stein theorem on sharp functions. See, for instance, [26,49], where the measure of the underlying space is clearly infinite. Here we deal with both the cases of a finite measure µ(X ) < ∞ and an infinite measure µ(X ) = ∞. We also present a different form of the Fefferman-Stein theorem. See Theorems 2.3 and 2.4 and Corollaries 2.6 and 2.7.The second objective, which is in fact a motivation of writing this paper, is to apply the generalized versions of the Fefferman-Stein theorem to establishing 2010 Mathematics Subject Classification. 35R05, 42B37, 35B45, 35K25, 35J48. Key words and phrases. sharp/maximal functions, elliptic and parabolic equations, weighted Sobolev spaces, measurable coefficients.H.

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“…The authors would like to thank the referees for their careful review as well as many valuable comments and suggestions, and for bringing our attention to the references [3,32,37,38].…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Instead of the above partitions one may use a significantly refined version of dyadic decompositions in [32]. DenoteX = n∈Z α Q n α .…”

Section: Generalized Fefferman-stein Theoremmentioning

confidence: 99%

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

Dong

Kim

2018

Trans. Amer. Math. Soc.

120

155

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“…Also, taking c 0 := 1, C 0 := 2A 0 and δ ≤ 10 −3 A −10 0 , we see that c 0 , C 0 and δ satisfy 12A 3 0 C 0 δ ≤ c 0 in [HK,Theorem 2.2]. By applying Hytönen and Kairema's construction ( [HK,Theorem 2.2]).…”

Section: Upper Bound Of Iterated Commutatormentioning

confidence: 99%

Product BMO, Little BMO, and Riesz Commutators in the Bessel Setting

Duong

et al. 2017

J Geom Anal

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Abstract. In this paper, we study the product BMO space, little bmo space and their connections with the corresponding commutators associated with Bessel operators studied by Weinstein, Huber, and by Muckenhoupt-Stein. We first prove that the product BMO space in the Bessel setting can be used to deduce the boundedness of the iterated commutators with the Bessel Riesz transforms. We next study the little bmo space in this Bessel setting and obtain the equivalent characterization of this space in terms of commutators, where the main tool that we develop is the characterization of the predual of little bmo and its weak factorizations. We further show that in analogy with the classical setting, the little bmo space is a proper subspace of the product BMO space. These extend the previous related results studied by Cotlar-Sadosky and Ferguson-Sadosky on the bidisc to the Bessel setting, where the usual analyticity and Fourier transform do not apply.

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“…In (X, µ) we choose a set of Christ's dyadic cubes [Chr90,HK12] Q using the scales {2 −n } n . The properties of Q that we will use are:…”

Section: A Local Approach To Fail Differentiabilitymentioning

confidence: 99%

“…Section 4 contains our proposal to implement the "gluing" part of the argument; this is more general than what we really use here, because we end up working with "cubical" tiles [Chr90,HK12]. Using other geometries for tiles might lead to a better understanding of the geometry of blow-ups; for example, Preiss and I discussed sometime ago what are essentially "long cylindrical" tiles to exclude factorizations of the form Y × R n in blow-ups of differentiability spaces.…”

Section: Introductionmentioning

confidence: 99%