ORCID Profile
0000-0003-0995-3054
Current Organisation
Macquarie University
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Lie Groups, Harmonic and Fourier Analysis | Pure Mathematics | Partial Differential Equations | Real and Complex Functions (incl. Several Variables)
Publisher: Springer Science and Business Media LLC
Date: 19-02-2011
Publisher: Elsevier BV
Date: 04-2019
Publisher: Elsevier BV
Date: 04-2021
Publisher: Wiley
Date: 06-06-2016
Publisher: Indiana University Mathematics Journal
Date: 2017
Publisher: Elsevier BV
Date: 04-2022
Publisher: Cambridge University Press (CUP)
Date: 10-2010
DOI: 10.1017/S144678871000159X
Abstract: We obtain an atomic decomposition for weighted Triebel–Lizorkin spaces on spaces of homogeneous type, using the area function, the discrete Calderón reproducing formula and discrete sequence spaces.
Publisher: Springer Science and Business Media LLC
Date: 29-12-2014
Publisher: Springer Science and Business Media LLC
Date: 17-05-2022
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2020
Publisher: Elsevier BV
Date: 12-2023
Publisher: Elsevier BV
Date: 09-2020
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2022
Publisher: Wiley
Date: 17-11-2016
DOI: 10.1002/MMA.4247
Publisher: Indiana University Mathematics Journal
Date: 2019
Publisher: Global Science Press
Date: 12-2009
Publisher: World Scientific Pub Co Pte Ltd
Date: 24-10-2021
DOI: 10.1142/S0219199720500625
Abstract: Let [Formula: see text], [Formula: see text] and [Formula: see text] be a matrix [Formula: see text] weight. In this paper, we introduce a version of variation [Formula: see text] for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the [Formula: see text]-boundedness of [Formula: see text] with norm [Formula: see text] by first proving a sparse domination of the variation of the scalar Calderón–Zygmund operator, and then providing a convex body sparse domination of the variation of the matrix Calderón–Zygmund operator. The key step here is a weak type estimate of a local grand maximal truncated operator with respect to the scalar Calderón–Zygmund operator.
Publisher: Elsevier BV
Date: 11-2021
Publisher: Springer Science and Business Media LLC
Date: 06-09-2017
Publisher: Elsevier BV
Date: 07-2019
Publisher: Springer Science and Business Media LLC
Date: 27-09-2017
Publisher: Elsevier BV
Date: 12-2016
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2016
Publisher: Elsevier BV
Date: 03-2013
Publisher: Springer Science and Business Media LLC
Date: 16-09-2018
Publisher: The Mathematical Society of the Republic of China
Date: 02-2010
Publisher: Elsevier BV
Date: 03-2020
Publisher: Springer Science and Business Media LLC
Date: 29-06-2023
Publisher: Springer Science and Business Media LLC
Date: 11-08-2012
Publisher: Elsevier BV
Date: 05-2011
Publisher: Springer Science and Business Media LLC
Date: 30-05-2012
Publisher: Hindawi Limited
Date: 2014
DOI: 10.1155/2014/265378
Abstract: Let ( X , d , μ ) be a space of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure μ satisfies only the doubling property. Adapting the recently developed randomized dyadic structures of X and applying orthonormal bases of L 2 ( X ) constructed recently by Auscher and Hytönen, we develop the Besov and Triebel-Lizorkin spaces on such a general setting. In this paper, we establish the wavelet characterizations and provide the dualities for these spaces. The results in this paper extend earlier related results with additional assumptions on the quasi-metric d and the measure μ to the full generality of the theory of these function spaces.
Publisher: American Mathematical Society (AMS)
Date: 17-06-2020
DOI: 10.1090/BPROC/48
Abstract: We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component. As ex les, we consider free groups on two generators, and show that these admit very simple polynomial embeddings in C N \\mathbb {C}^N that induce their CR structure.
Publisher: Cellule MathDoc/CEDRAM
Date: 2018
DOI: 10.5802/AIF.3153
Publisher: Springer Science and Business Media LLC
Date: 24-10-2020
Publisher: American Mathematical Society (AMS)
Date: 09-2022
DOI: 10.1090/MEMO/1373
Abstract: In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood–Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel–Pólya inequalities for flag quadratic forms (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure.
Publisher: American Mathematical Society (AMS)
Date: 22-11-2023
DOI: 10.1090/PROC/15296
Abstract: We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.
Publisher: Elsevier BV
Date: 2011
Publisher: Michigan Mathematical Journal
Date: 03-2023
DOI: 10.1307/MMJ/20205900
Publisher: Springer Science and Business Media LLC
Date: 16-03-2021
Publisher: Springer Science and Business Media LLC
Date: 11-11-2021
Publisher: Elsevier BV
Date: 2021
Publisher: Springer Science and Business Media LLC
Date: 24-02-2023
DOI: 10.1007/S00041-023-09995-1
Abstract: We study commutators with the Riesz transforms on the Heisenberg group $${\\mathbb {H}}^{n}$$ H n . The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson–Wolff and Rochberg–Semmes. The method in proof bypasses the use of Fourier analysis, allowing us to address not just the Riesz transforms, but also the Cauchy–Szegő projection and second order Riesz transforms on $${\\mathbb {H}}^{n}$$ H n among other settings.
Publisher: Springer Science and Business Media LLC
Date: 14-11-2018
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2021
Publisher: Springer Science and Business Media LLC
Date: 24-08-2022
Publisher: American Mathematical Society (AMS)
Date: 25-04-2013
Publisher: Elsevier BV
Date: 10-2016
Publisher: Indiana University Mathematics Journal
Date: 2021
Publisher: Elsevier BV
Date: 11-2013
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2020
Publisher: Mathematical Society of Japan (Project Euclid)
Date: 25-01-2022
Publisher: Springer Science and Business Media LLC
Date: 20-06-2021
Publisher: Indiana University Mathematics Journal
Date: 2021
Publisher: Springer Science and Business Media LLC
Date: 06-2018
Publisher: Walter de Gruyter GmbH
Date: 2011
Publisher: Elsevier BV
Date: 12-2018
Publisher: Springer Science and Business Media LLC
Date: 25-02-2017
Publisher: Springer Science and Business Media LLC
Date: 02-10-2017
Publisher: Canadian Mathematical Society
Date: 09-2017
Abstract: This paper provides a constructive proof of the weak factorization of the classical Hardy space H 1 (ℝ n ) in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of BMO(ℝ n ) (the dual of H 1 (ℝ n )) via commutators of the multilinear Riesz transforms.
Publisher: The Mathematical Society of the Republic of China
Date: 04-2019
DOI: 10.11650/TJM/181203
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2022
Publisher: Elsevier BV
Date: 07-2018
Publisher: Springer Science and Business Media LLC
Date: 06-04-2023
DOI: 10.1007/S11118-023-10072-X
Abstract: We characterize the Hilbert–Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class S p result for dyadic paraproducts, where $1 p \\infty $ 1 p ∞ . We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.
Publisher: American Mathematical Society (AMS)
Date: 24-07-2012
DOI: 10.1090/S0002-9947-2012-05638-8
Abstract: This paper is inspired by the work of Nagel and Stein in which the L p L^p ( 1 p ∞ ) (1 \\infty ) theory has been developed in the setting of the product Carnot-Carathéodory spaces M ~ = M 1 × ⋯ × M n \\widetilde {M}=M_1\\times \\cdots \\times M_n formed by vector fields satisfying Hörmander’s finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product B M O BMO space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.
Publisher: Indiana University Mathematics Journal
Date: 2020
Publisher: Springer Science and Business Media LLC
Date: 03-06-2020
Publisher: Elsevier BV
Date: 09-2019
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 2015
DOI: 10.4171/RMI/853
Publisher: Springer Science and Business Media LLC
Date: 20-11-2018
Publisher: American Mathematical Society (AMS)
Date: 22-04-2011
DOI: 10.1090/S0002-9939-2011-10852-9
Abstract: This article deals with the characterizations of Hardy space H 1 H^1 on R n × R m \\mathbb {R}^n\\times \\mathbb {R}^m using different norms on distinct variables. This result can be applied to the boundedness of certain operators on H 1 ( R n × R m ) H^1(\\mathbb {R}^n\\times \\mathbb {R}^m) .
Publisher: Elsevier BV
Date: 10-2023
Publisher: Springer Science and Business Media LLC
Date: 26-05-2017
Publisher: Canadian Mathematical Society
Date: 09-01-2019
Abstract: The Marcinkiewicz multipliers are $L^{p}$ bounded for $1 \\infty$ on the Heisenberg group $\\mathbb{H}^{n}\\simeq \\mathbb{C}^{n}\\times \\mathbb{R}$ (Müller, Ricci, and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\\mathbb{C}^{n}\\times \\mathbb{R}$ , while there is no two parameter group of automorphic dilations on $\\mathbb{H}^{n}$ . The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\\mathbb{H}^{n}\\simeq \\mathbb{C}^{n}\\times \\mathbb{R}$ that is, in a sense, intermediate between that of the classical Lipschitz space on the Heisenberg group $\\mathbb{H}^{n}$ and the product Lipschitz space on $\\mathbb{C}^{n}\\times \\mathbb{R}$ . We characterize this flag Lipschitz space via the Littlewood–Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.
Publisher: MitoFit Preprint Archives
Date: 2019
Publisher: Springer Science and Business Media LLC
Date: 23-08-2018
Publisher: Elsevier BV
Date: 2024
Publisher: Mathematical Society of Japan (Project Euclid)
Date: 2019
Publisher: Elsevier BV
Date: 02-2017
Publisher: Springer Science and Business Media LLC
Date: 22-12-2022
Publisher: Springer Science and Business Media LLC
Date: 05-11-2016
Publisher: Cellule MathDoc/CEDRAM
Date: 15-02-2022
DOI: 10.5802/CRMATH.265
Publisher: Springer Science and Business Media LLC
Date: 17-05-2023
Publisher: Walter de Gruyter GmbH
Date: 2020
Abstract: In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type ( X , d , µ ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [ b , T ] is bounded on the weighted Morrey space L ω p , k ( X ) L_\\omega ^{p,k}\\left( X \\right) with κ ∈ (0, 1) and ω ∈ A p ( X ), 1 p ∞, if and only if b is in the BMO space. We also prove that the commutator [ b , T ] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ .
Publisher: Springer Science and Business Media LLC
Date: 20-03-2016
Publisher: Elsevier BV
Date: 06-2017
Publisher: Springer Science and Business Media LLC
Date: 23-03-2023
DOI: 10.1007/S12220-023-01239-4
Abstract: We establish the maximal operator, Cotlar’s inequality and pointwise convergence in the Dunkl setting for the (nonconvolution type) Dunkl–Calderón–Zygmund operators introduced recently in Tan et al. ( bs/2204.01886 ). The fundamental geometry of the Dunkl setting involves two nonequivalent metrics: the Euclidean metric and the Dunkl metric deduced by finite reflection groups, and hence the classical methods do not apply directly. The key idea is to introduce truncated singular integrals and the maximal singular integrals by the Dunkl metric and the Euclidean metric. We show that these two kind of truncated singular integrals are dominated by the Hardy–Littlewood maximal function, which yields the Cotlar’s inequalities and hence the boundedness of maximal Dunkl–Calderón–Zygmund operators. Further, as applications, two equivalent pointwise convergences for Dunkl–Calderón–Zygmund operators are obtained.
Publisher: Elsevier BV
Date: 02-2019
Publisher: Elsevier BV
Date: 04-2018
Publisher: Indiana University Mathematics Journal
Date: 2009
Publisher: Springer Science and Business Media LLC
Date: 25-08-2020
Publisher: Mathematical Sciences Publishers
Date: 2019
Start Date: 02-2016
End Date: 06-2020
Amount: $363,100.00
Funder: Australian Research Council
View Funded ActivityStart Date: 05-2022
End Date: 05-2025
Amount: $375,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2017
End Date: 11-2020
Amount: $350,000.00
Funder: Australian Research Council
View Funded Activity