Littlewood–Paley Theory and the T(1) Theorem with Non-doubling Measures

By Xavier Tolsa. 2000

139 citations recorded by the source corpus.

Let μ be a Radon measure on Rd which may be non-doubling. The only condition that μ must satisfy is μ(B(x, r))⩽Crn, for all x∈Rd, r>0, and for some fixed 0<n⩽d. In this paper, Littlewood–Paley theory for functions in Lp(μ) is developed. One of the main difficulties to be solved is the construction of “reasonable” approximations of the identity in order to obtain a Calderon type reproducing formula. Moreover, it is shown that the T(1) theorem for n-dimensional Calderon–Zygmund operators, without doubling assumptions, can be proved using the Littlewood–Paley type decomposition that is obtained for functions in L2(μ), as in the classical case of homogeneous spaces.}

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cac3d2950b95e8c9109548fd56a1fe1a0310e515

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Citations in the corpus

Multilinear Commutators of Singular Integrals with Non Doubling Measures (2005, background): “During the last several years, considerable attention has been paid to the study of boundedness of singular integrals with non doubling measures; see [2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15 , 16].”
Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, methodology): “Tolsa’s technique seems reasonable as far as we know how to prove the weak type (1,1) inequality.”
SHARP MAXIMAL INEQUALITIES AND COMMUTATORS ON MORREY SPACES WITH NON-DOUBLING MEASURES (2007, unknown): “He also gave his H1(μ) space in terms of the grand maximal operator [15].”
BOUNDEDNESS OF MULTILINEAR COMMUTATORS OF CALDERÓN-ZYGMUND OPERATORS ON ORLICZ SPACES OVER NON-HOMOGENEOUS SPACES (2012, methodology): “However, in recent years, many results in the classical theory of Calderón-Zygmund operators on R have been proved still valid with the Lebesgue measure m replaced by a Radon measure μ satisfying the polynomial growth condition (see, for example, [22, 23, 24, 25, 26, 27, 17, 3, 7, 8]).”
Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces (2010, background): “However, recently, many results on the classical Calderón-Zygmund theory have been proved still valid if the measure doubling condition is replaced by a less demanding condition; see, for example, [13, 16, 17, 14, 18, 12, 2] and the references therein.”
GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES (2013, methodology): “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”
Morrey Spaces for Non–doubling Measures (2005, unknown): “[2, 3]). However, recently it has been shown that many results in this theory also hold without the doubling assumption; see [4, 5 , 6] and many other literatures.”
Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces for nondoubling measures (2009, methodology): “All the definitions are introduced by Tolsa (see [26, 27, 28]).”
The Hardy space H1 on non-homogeneous metric spaces (2010, methodology): “We now recall the coefficients δ(B, S) for all balls B and S which were introduced in [6] as analogues of Tolsa’s numbers KQ,R from [14]; see also [15, 17].”