Systems of dyadic cubes in a doubling metric space

By Tuomas Hytönen, Anna Kairema. 2010

332 citations recorded by the source corpus.

A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.

View Paper: https://www.semanticscholar.org/paper/fecfb23e6f50c47bf6ea30f9cbc5e73971923682

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Citations in the corpus

Multilinear analysis on metric spaces (2014, background): “[1, 67, 68] constructed a randomized dyadic structure by only assuming that the underlying metric space is geometrically doubling.”
Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “We ﬁrst recall several results about the construction of adjacent dyadic systems (see [Christ 1990; Sawyer and Wheeden 1992; Hytönen and Kairema 2012] for more details).”
On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights (2016, background): “Instead of the above partitions one may use a significantly refined version of dyadic decompositions in [32].”
Noncommutative maximal ergodic inequalities associated with doubling conditions (2017, methodology): “ical argument via covering lemmas does not seem to ﬁt into this operator-valued setting. Instead, our approach is based on the study of random dyadic systems by Naor, Tao [NT10] and Hyt¨onen, Kaimera [HK12]. The key idea is to relate the desired inequality to noncommutative martingales, and to use t”
Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases (2015, methodology): “ssumptions are made on the quasi-metric and the measure, or the product case is not covered, or both. The paper is organized as follows. In Section 2 we brieﬂy recall the systems of dyadic cubes from [HK] and the orthonormal bases from [AH] on spaces ofhomogeneous type in the sense of Coifman and We”
Orthonormal bases of regular wavelets in spaces of homogeneous type (2011, methodology): “In [19, 21], a systematic method of constructing several Christ-type families of dyadic cubes Qα(ω) was introduced, where ω belongs to a parameter space Ω.”
Two Weight Commutators on Spaces of Homogeneous Type and Applications (2018, methodology): “We recall from [26] the following construction.”