Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces
By Tuomas Hytönen, Henri Martikainen. 2009
164 citations recorded by the source corpus.
We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound μ(B(x,r))≤Crd. Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/763e29fee609895990878d5bbbb49159cc35a530
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- $L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets (2013, methodology): “Analogous T (b) theorems were obtained by Hÿtonen [50] (in Euclidean space) and by Hÿtonen and Martikainen [51] (in a metric measure space).”
- Systems of dyadic cubes in a doubling metric space (2010, methodology): “n to the details of the proofs. As in Rn, where the open (2−k((0,1)n + m)), half-open (2−k([0,1)n+m)) and closed (2−k([0,1]n+m)) dyadic cubes each serve their different purpose, we also present (as in [12]) a unified construction of three related systems of open, half-open and closed cubes. While this”
- Representation of bi-parameter singular integrals by dyadic operators (2011, unknown): “Modifying the metric randomization by Hytönen and the author [9] these representation theorems were lifted to the generality of metric spaces by Nazarov, Reznikov and Volberg [14].”
- 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.”
- A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, unknown): “(And after the submission of this paper, this framework has proven to be amenable even for the more general goal of Tb theorems for CalderónZygmund operators in my joint work with Martikainen [12].”
- Orthonormal bases of regular wavelets in spaces of homogeneous type (2011, methodology): “iscovered that several aspects of harmonic analysis can even be pushed beyond the setting of doubling measures; see e.g. [33, 40, 41] for some of the pioneering developments in the context of Rd, and [20, 21] for recent developments in abstract quasi-metric spaces. We do not address the non-doubling”
- The sharp weighted bound for general Calderón-Zygmund operators (2010, background): “particular, the dyadic cubes in this generality have already been constructed by Christ [4], and the required randomisation of this construction was recently carried out by Martikainen and the author [12]. The present arguments also made use of some specific symmetries of the Euclidean space, especia”
- The sharp weighted bound for general Calderon-Zygmund operators (2010, unknown): “In particular, the dyadic cubes in this generality have already been constructed by Christ [4], and the required randomisation of this construction was recently carried out by Martikainen and the author [14].”
- Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces (2010, methodology): “Later, Hytönen and Martikainen [10] further established a version of the T(b) theorem for Calderón–Zygmund operators in such spaces.”
- Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces (2010, methodology): “ips B2n with the regular quasi-distance, see [Tch, Lemma 2.6], d(x,y) := |x|−|y| + 1− x ·y |x||y| . Finally, the set H related to the exceptional balls is now the open unit ball B2n. It was proved in [HyM] that the kernel K(x,y) defined by (4) satisfies (2) and (3). 3 The RBMO spaces 3.1 Definition of ”
- GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES (2013, background): “Recently, many classical results have been proved still valid if the underlying spaces are replaced by the non-homogeneous metric measure spaces (see, for example, [18, 22, 2, 19, 20, 21, 25, 8, 24]).”