TheTb-theorem on non-homogeneous spaces

By Fedor Nazarov, Sergei Treil, Alexander Volberg. 2003

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0 Introduction: main objects and results 3 0.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.2 An application of T1-heorem: electric intensity capacity . . . . . . . . . . . . 7 0.3 How to interpret Calderon–Zygmund operator T? . . . . . . . . . . . . . . . 9 0.3.1 Bilinear form is defined on Lipschitz functions . . . . . . . . . . . . . 10 0.3.2 Bilinear form is defined for smooth functions . . . . . . . . . . . . . . 11 0.3.3 Apriori boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.4 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/800da6d5b36319296f010f6d852db2b65fe7984d

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Citations in the corpus

GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES (2013, background): “1 Introduction During the past ten to fifteen years, considerable attention has been paid to the study of the classical theory of harmonic analysis on Euclidean spaces with non-doubling measures only satisfying the polynomial growth condition (see, for example, [11, 10, 35, 36, 37, 38, 39, 40, 5, 27”
H^1 and BMO for certain locally doubling metric measure spaces of finite measure (2008, background): “Finally, we recall that Hardy spaces and spaces of functions of bounded mean oscillation have recently been studied on various nondoubling metric measure spaces [14, 18, 19, 20].”
A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis (2010, methodology): “ {0,1}Z, deﬁne the dyadic grid Dβ to be the collection of intervals Dβ = ( 2n [0,1)+k + X i<n 2i−nβ i !) n∈Z,k∈Z This parametrization of dyadic grids appears explicitly in [Hyt], and implicitly in [NTV2] section 9.1. Place the usual uniform probability measure Pon the space {0,1}Z, explicitly P(β”
The sharp weighted bound for general Calderon-Zygmund operators (2010, methodology): “This is once again provided by the proof of a T (1) theorem — this time, the one for nonhomogeneous spaces due to Nazarov, Treil and Volberg [22].”
A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, methodology): “In the first setting, the inequality is due to Tolsa [20], and reproven in [18].”
Uniform sparse domination of singular integrals via dyadic shifts (2016, methodology): “We were introduced to definition (3.1) in the context of sparse domination by S. Treil, who, jointly with Nazarov, Petermichl and Volberg, announced the following result ([22], with full proof appearing in [21]): for a standard real-valued Calderón-Zygmund kernel operator and for f in a suitable den”
Sharp weighted bounds for the q‐variation of singular integrals (2012, methodology): “was an elaboration of the proof of the T (1) theorem of David–Journé [4], or more precisely, of the non-homogeneous version due to Nazarov–Treil–Volberg [14], and in effect reproved the boundedness of T as a by-product.”
Systems of dyadic cubes in a doubling metric space (2010, background): “These random dyadic systems have been instrumental in the development of the non-doubling theory of singular integrals [23, 24] and its applications to analytic capacity [27, 28] as well as sharp one-weight [9, 13] and two-weight [16, 25] inequalities for classical singular integrals.”
The Hardy space H1 on non-homogeneous metric spaces (2010, methodology): “Some highlights of this theory are the introduction of the atomic Hardy space H 1(µ) and its dual space RBMO (µ) (the space of functions with regularized bounded mean oscillation) by Tolsa [14], the proof of the T b theorem by Nazarov, Treil and Volberg [12] and the solution of the Painlevé problem”
The Bellman function, the two-weight Hilbert transform, and embeddings of the model spacesKθ (2002, background): “This brings us to the circle of questions of nonhomogeneous harmonic analysis explored in [22], [23], [25], [26], [28], [42]-[43], [15], [12]-[14], especially in what concerns various Tb theorems as in [ 25 ], [26], [7].”
The A_2 theorem: Remarks and complements (2012, methodology): “2) a reduction of the operator T to dyadic model operators called shifts Sk via a probabilistic argument inspired by the work of Nazarov, Treil and Volberg on nondoubling harmonic analysis [22]; and a subtle multi-step (“corona”) decomposition, elaborating on earlier work of Lacey, Petermichl and Re”
Representation of bi-parameter singular integrals by dyadic operators (2011, positive): “ product, a pretty nice form of the product space T1 theorem. In this paper we bring the superbly useful machinery of non-homogeneous analysis pioneered by Nazarov, Treil and Volberg (see for example [15]) to the contextofbi-parametertheory. Theuseofnon-homogeneousanalysisgivesadditional decayfor ce”
A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, methodology): “In the first setting, the inequality is due to Tolsa [20], and reproven in [18].”
Random Martingales and localization of maximal inequalities (2009, background): “[39] F. Nazarov, S. Treil, and A. Volberg.”
Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, methodology): “Recall that Littlewood-Paley theory for nondoubling measures was a corner stone through the T1 theorem by Nazarov, Treil and Volberg [ 22 ] and Tolsa [38].”