ORCID Profile
0000-0002-9855-3249
Current Organisation
Macquarie University
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Lie Groups, Harmonic and Fourier Analysis | Pure Mathematics | Partial Differential Equations | Operator Algebras and Functional Analysis | Algebraic and Differential Geometry
Publisher: Springer Science and Business Media LLC
Date: 24-10-2007
Publisher: Oxford University Press (OUP)
Date: 21-01-2018
DOI: 10.1093/IMRN/RNW323
Publisher: Springer Science and Business Media LLC
Date: 23-04-2010
Publisher: International Press of Boston
Date: 2004
Publisher: Springer Science and Business Media LLC
Date: 07-12-2008
Publisher: Springer Science and Business Media LLC
Date: 20-09-2013
Publisher: Springer Science and Business Media LLC
Date: 27-10-2016
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2021
Publisher: Elsevier BV
Date: 12-2016
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 04-2007
Publisher: Birkhäuser Basel
Date: 2006
Publisher: Springer Science and Business Media LLC
Date: 07-2016
Publisher: Springer Science and Business Media LLC
Date: 08-04-2020
Publisher: Wiley
Date: 24-11-2008
DOI: 10.1112/PLMS/PDM050
Publisher: Indiana University Mathematics Journal
Date: 2009
Publisher: American Mathematical Society (AMS)
Date: 13-07-2022
DOI: 10.1090/TRAN/8695
Abstract: We investigate the Hardy space H L 1 H^1_L associated with a self-adjoint operator L L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L L -harmonic non-negative function h h such that the semigroup exp ( − t L ) \\exp (-tL) , after applying the Doob transform related to h h , satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H L 1 H^1_L in terms of a simple atomic decomposition associated with the L L -harmonic function h h . Our approach also yields a natural characterisation of the B M O BMO -type space corresponding to the operator L L and dual to H L 1 H^1_L in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in R n {\\mathbb {R}^n} , Schrödinger operators with certain potentials, and Bessel operators.
Publisher: Walter de Gruyter GmbH
Date: 05-10-2018
Abstract: On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.
Publisher: American Mathematical Society (AMS)
Date: 02-04-2013
DOI: 10.1090/S0002-9947-2013-05849-7
Abstract: Let M ∘ M^\\circ be a complete noncompact manifold and g g an asymptotically conic Riemaniann metric on M ∘ M^\\circ , in the sense that M ∘ M^\\circ compactifies to a manifold with boundary M M in such a way that g g becomes a scattering metric on M M . Let Δ \\Delta be the positive Laplacian associated to g g , and P = Δ + V P = \\Delta + V , where V V is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure d E ( λ ) = ( λ / π i ) ( R ( λ + i 0 ) − R ( λ − i 0 ) ) dE(\\lambda ) = (\\lambda /\\pi i) \\big ( R(\\lambda +i0) - R(\\lambda - i0) \\big ) of P + 1 / 2 P_+^{1/2} , where R ( λ ) = ( P − λ 2 ) − 1 R(\\lambda ) = (P - \\lambda ^2)^{-1} , as λ → 0 \\lambda \\to 0 , in a manner similar to that done by the second author and Vasy (2001) and by the first two authors (2008, 2009). The main result is that the spectral measure has a simple, ‘conormal-Legendrian’ singularity structure on a space which was introduced in the 2008 work of the first two authors and is obtained from M 2 × [ 0 , λ 0 ) M^2 \\times [0, \\lambda _0) by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators cos ( t P + ) \\cos (t \\sqrt {P_+}) and sin ( t P + ) / P + \\sin (t \\sqrt {P_+})/\\sqrt {P_+} , and the Schrödinger propagator e i t P + e^{itP_+} , as t → ∞ t \\to \\infty . In particular, we prove the analogue of Price’s law for odd-dimensional asymptotically conic manifolds. In future articles, this result on the spectral measure will be used to (i) prove restriction and spectral multiplier estimates on asymptotically conic manifolds, and (ii) prove long-time dispersion and Strichartz estimates for solutions of the Schrödinger equation on M M , provided M M is nontrapping.
Publisher: Mathematical Sciences Publishers
Date: 21-08-2013
Publisher: Elsevier BV
Date: 06-2023
Publisher: Informa UK Limited
Date: 06-06-2019
Publisher: Elsevier BV
Date: 2014
Publisher: Springer Science and Business Media LLC
Date: 11-2001
Publisher: IEEE
Date: 2005
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2010
DOI: 10.4064/CM118-2-20
Publisher: Elsevier BV
Date: 02-1999
Publisher: Springer Science and Business Media LLC
Date: 12-03-2014
Publisher: Elsevier BV
Date: 2015
Publisher: Elsevier BV
Date: 02-2011
Publisher: Indiana University Mathematics Journal
Date: 2009
Publisher: Springer Science and Business Media LLC
Date: 07-03-2017
Publisher: Wiley
Date: 24-04-2007
DOI: 10.1112/PLMS/PDL017
Publisher: American Mathematical Society (AMS)
Date: 02-03-2018
DOI: 10.1090/TRAN/8024
Abstract: We investigate L p L^p boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic-type. Assuming the finite speed of propagation for the associated wave operator, from the restriction-type estimates we establish the sharp L p L^p boundedness of the maximal Bochner-Riesz means for the elliptic operators. As applications, we obtain the sharp L p L^p maximal bounds for the Schrödinger operators on asymptotically conic manifolds, elliptic operators on compact manifolds, or the Hermite operator and its perturbations on R n \\mathbb {R}^n .
Publisher: American Mathematical Society (AMS)
Date: 02-2011
Publisher: Elsevier BV
Date: 11-2020
Publisher: Elsevier BV
Date: 05-2020
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2011
DOI: 10.4064/SM203-1-5
Publisher: Elsevier BV
Date: 12-2002
Publisher: Cambridge University Press (CUP)
Date: 06-2011
DOI: 10.1017/S1446788711001315
Abstract: Let S be a sub-Markovian semigroup on L 2 (ℝ d ) generated by a self-adjoint, second-order, ergence-form, elliptic operator H with W 1, ∞ (ℝ d ) coefficients c kl , and let Ω be an open subset of ℝ d . We prove that if either C ∞ c (ℝ d ) is a core of the semigroup generator of the consistent semigroup on L p (ℝ d ) for some p ∈[1, ∞ ] or Ω has a locally Lipschitz boundary, then S leaves L 2 (Ω) invariant if and only if it is invariant under the flows generated by the vector fields ∑ d l =1 c kl ∂ l for all k . Further, for all p ∈[1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various ex les of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.
Start Date: 02-2013
End Date: 12-2017
Amount: $330,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2011
End Date: 12-2015
Amount: $520,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 05-2016
End Date: 12-2021
Amount: $445,118.00
Funder: Australian Research Council
View Funded ActivityStart Date: 04-2020
End Date: 12-2024
Amount: $420,000.00
Funder: Australian Research Council
View Funded Activity