ORCID Profile
0000-0003-2793-296X
Current Organisation
Macquarie University
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Lie Groups, Harmonic and Fourier Analysis | Pure Mathematics | Real and Complex Functions (incl. Several Variables) | Partial Differential Equations
Publisher: Elsevier BV
Date: 04-2021
Publisher: American Mathematical Society (AMS)
Date: 13-01-2020
DOI: 10.1090/PROC/14845
Publisher: The Mathematical Society of the Republic of China
Date: 07-2013
Publisher: Informa UK Limited
Date: 18-03-2010
Publisher: Springer Science and Business Media LLC
Date: 02-08-2016
Publisher: Springer Science and Business Media LLC
Date: 15-06-2019
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 17-12-2019
DOI: 10.4171/RMI/1136
Publisher: Elsevier BV
Date: 2013
Publisher: Mathematical Institute, Tohoku University
Date: 2014
Publisher: Cambridge University Press (CUP)
Date: 09-2011
Abstract: Let L be a nonnegative self-adjoint operator on L 2 (X ), where X is a space of homogeneous type. Assume that L generates an analytic semigroup e –tl whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier F(L ) is bounded on for 0 p 1, the Hardy space associated to operator L , when F is a suitable function.
Publisher: Springer Science and Business Media LLC
Date: 04-03-2020
Publisher: Elsevier BV
Date: 11-2016
Publisher: Elsevier BV
Date: 03-2014
Publisher: Springer Science and Business Media LLC
Date: 08-01-2021
Publisher: Elsevier BV
Date: 10-2016
Publisher: Mathematical Society of Japan (Project Euclid)
Date: 23-04-2021
Publisher: Elsevier BV
Date: 12-2022
Publisher: Pleiades Publishing Ltd
Date: 12-2010
Publisher: Springer Science and Business Media LLC
Date: 12-05-2009
Publisher: Oxford University Press (OUP)
Date: 23-01-2021
DOI: 10.1093/IMRN/RNAA002
Abstract: In this paper, by using a new approach, we prove regularity estimates for the solution to the non ergence parabolic equation on generalized Orlicz spaces. Our approach can be viewed as a combination of representation theorems in partial differential equations and harmonic analysis techniques.
Publisher: Springer Science and Business Media LLC
Date: 07-06-2018
Publisher: Springer Science and Business Media LLC
Date: 27-05-2006
Publisher: Springer Science and Business Media LLC
Date: 28-07-2023
DOI: 10.1007/S00041-023-10032-4
Abstract: Let $$(X, d, \\mu )$$ ( X , d , μ ) be a space of homogeneous type. Let L be a nonnegative self-adjoint operator on $$L^2(X)$$ L 2 ( X ) satisfying certain conditions on the heat kernel estimates which are motivated from the heat kernel of the Schrödinger operator on $$\\mathbb {R}^n$$ R n . The main aim of this paper is to prove a new atomic decomposition for the Besov space $$\\dot{B}^{0, L}_{1,1}(X)$$ B ˙ 1 , 1 0 , L ( X ) associated with the operator L . As a consequence, we prove the boundedness of the Riesz transform associated with L on the Besov space $$\\dot{B}^{0, L}_{1,1}(X)$$ B ˙ 1 , 1 0 , L ( X ) .
Publisher: Elsevier BV
Date: 2021
Publisher: American Mathematical Society (AMS)
Date: 30-06-2022
DOI: 10.1090/PROC/16017
Abstract: Let ( X , d , μ ) (X,d, \mu ) be an Ahlfors n n -regular metric measure space. Let L \mathcal {L} be a non-negative self-adjoint operator on L 2 ( X ) L^2(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F ( L ) F(\mathcal {L}) satisfy an appropriate weighted L 2 L^2 estimate. By the spectral theory, we can define the imaginary power operator L i s , s ∈ R \mathcal {L}^{is}, s\in \mathbb R , which is bounded on L 2 ( X ) L^2(X) . The main aim of this paper is to prove that for any p ∈ ( 0 , ∞ ) p \in (0,\infty ) , ‖ L i s f ‖ H L p ( X ) ≤ C ( 1 + | s | ) n | 1 / p − 1 / 2 | ‖ f ‖ H L p ( X ) , s ∈ R , \begin{equation*} \big \|\mathcal {L}^{is} f\big \|_{H^p_{\mathcal {L}}(X)} \leq C (1+|s|)^{n|1 -1/2|} \|f\|_{H^p_{\mathcal {L}}(X)}, \quad s \in \mathbb {R}, \end{equation*} where H L p ( X ) H^p_\mathcal {L}(X) is the Hardy space associated to L \mathcal {L} , and C C is a constant independent of s s . Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on R n \mathbb {R}^n with n ≥ 2 n\ge 2 .
Publisher: Springer Science and Business Media LLC
Date: 08-2022
DOI: 10.1007/S00041-022-09964-0
Abstract: Let X be a space of homogeneous type with the doubling order n . Let L be a nonnegative self-adjoint operator on $$L^2(X)$$ L 2 ( X ) and suppose that the kernel of $$e^{-tL}$$ e - t L satisfies a Gaussian upper bound. This paper shows that for $$0 \le 1$$ 0 p ≤ 1 and $$s=n(1 -1/2)$$ s = n ( 1 / p - 1 / 2 ) , $$\begin{aligned}\Vert (I+L)^{-s}e^{itL}f\Vert _{H^p_L(X)} \lesssim (1+|t|)^{s}\Vert f\Vert _{H^p_L(X)} \end{aligned}$$ ‖ ( I + L ) - s e itL f ‖ H L p ( X ) ≲ ( 1 + | t | ) s ‖ f ‖ H L p ( X ) for all $$t\in {\mathbb {R}}$$ t ∈ R , where $$H^p_L(X)$$ H L p ( X ) is the Hardy space associated to L . This recovers the classical results in the particular case when $$L=-\Delta $$ L = - Δ and extends a number of known results.
Publisher: World Scientific Pub Co Pte Lt
Date: 10-03-2021
DOI: 10.1142/S0219199720500145
Abstract: This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non- ergence form with BMO coefficients in a [Formula: see text] domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non- ergence elliptic equations and the domination technique by sparse operators in harmonic analysis.
Publisher: Springer Science and Business Media LLC
Date: 13-12-2014
Publisher: Informa UK Limited
Date: 12-2010
Publisher: Springer Science and Business Media LLC
Date: 27-05-2022
Publisher: Springer Science and Business Media LLC
Date: 16-03-2021
Publisher: American Mathematical Society (AMS)
Date: 17-02-2014
Publisher: Springer Science and Business Media LLC
Date: 08-11-2017
Publisher: American Mathematical Society (AMS)
Date: 17-05-2017
DOI: 10.1090/TRAN/6745
Abstract: Consider the space X = ( 0 , ∞ ) X=(0,\\infty ) equipped with the Euclidean distance and the measure d μ α ( x ) = x α d x d\\mu _\\alpha (x)=x^{\\alpha }dx where α ∈ ( − 1 , ∞ ) \\alpha \\in (-1,\\infty ) is a fixed constant and d x dx is the Lebesgue measure. Consider the Laguerre operator L = − d 2 d x 2 − α x d d x + x 2 \\displaystyle L=-\\frac {d^2}{dx^2} -\\frac {\\alpha }{x}\\frac {d}{dx}+x^2 on X X . The aim of this article is threefold. Firstly, we establish a Calderón reproducing formula using a suitable distribution of the Laguerre operator. Secondly, we study certain properties of the Laguerre operator such as a Harnack type inequality on the solutions and subsolutions of Laplace equations associated to Laguerre operators. Thirdly, we establish the theory of the weighted homogeneous Besov and Triebel-Lizorkin spaces associated to the Laguerre operator. We define the weighted homogeneous Besov and Triebel-Lizorkin spaces by the square functions of the Laguerre operator, then show that these spaces have an atomic decomposition. We then study the fractional powers L − γ , γ 0 L^{-\\gamma }, \\gamma , and show that these operators map boundedly from one weighted Besov space (or one weighted Triebel-Lizorkin space) to another suitable weighted Besov space (or weighted Triebel-Lizorkin space). We also show that in particular cases of the indices, our new weighted Besov and Triebel-Lizorkin spaces coincide with the (expected) weighted Hardy spaces, the weighted L p L^p spaces or the weighted Sobolev spaces in Laguerre settings.
Publisher: Walter de Gruyter GmbH
Date: 07-02-2013
Abstract: Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL qL) off-diagonal estimates on balls, where pL ∊ [1 2) and qL ∊ (2 ∞]. Let φ : X × [0 ∞) → [0 ∞) be a function such that φ (x ·) is an Orlicz function, φ(· t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0 ] and φ(· t) satisfies the uniformly reverse Hölder inequality of order (qL/l(φ))′, where (qL/l(φ))′ denotes the conjugate exponent of qL/l(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ,L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ,L(ℝn) and the classical Musielak-Orlicz-Hardy space Hv(ℝn) is given. Moreover, for the Musielak-Orlicz-Hardy space Hφ,L(ℝn) associated with the second order elliptic operator in ergence form on ℝn or the Schrödinger operator L := −Δ + V with 0 ≤ V ∊ L1loc(ℝn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(ℝn) to the Musielak-Orlicz space Lφ(ℝn) when i(φ) ∊ (0 1], from Hφ,L(ℝn) to Hφ(ℝn) when i(φ) ∊ ( 1], and from Hφ,L(ℝn) to the weak Musielak-Orlicz-Hardy space WHφ(ℝn) when i(φ)= is attainable and φ(· t) ∊ A1(X), where i(φ) denotes the uniformly critical lower type index of φ
Publisher: Springer Science and Business Media LLC
Date: 13-06-2022
Publisher: Elsevier BV
Date: 09-2014
Publisher: Elsevier BV
Date: 03-2019
Publisher: Springer Science and Business Media LLC
Date: 05-2019
Publisher: Elsevier BV
Date: 2011
Publisher: Hindawi Limited
Date: 2008
DOI: 10.1155/2008/124269
Abstract: We study the robustness of strong stability of the homogeneous difference equation via the concept of strong stability radii: complex, real and positive radii in this paper. We also show that in the case of positive systems, these radii coincide. Finally, a simple ex le is given.
Publisher: Springer Science and Business Media LLC
Date: 07-11-2020
Publisher: Wiley
Date: 27-03-2009
DOI: 10.1002/RNC.1414
Publisher: Springer Science and Business Media LLC
Date: 26-12-2022
Publisher: Springer Science and Business Media LLC
Date: 23-02-2022
DOI: 10.1007/S00030-022-00751-W
Abstract: In this paper, we prove new estimates on temporal-spatial decays in $$L^1$$ L 1 for solutions to the Stokes equations in the half spaces. Our approach is based on the Ukai’s solution formula of the Stokes equations and heat kernel estimates.
Publisher: Elsevier BV
Date: 05-2020
Publisher: Oxford University Press (OUP)
Date: 03-2022
DOI: 10.1093/IMRN/RNAC037
Abstract: Let $\mathcal {H}=-\Delta + |x|^2$ be the Hermite operator on $\mathbb R^n$ with $n\ge 2$. In this paper, we prove the boundedness of Schrödinger groups of fractional powers of $\mathcal {H}$ on Lebesgue and Hardy spaces. More precisely, we prove that (a) for $p \in (1,\infty )$, $\gamma& $ and $\beta /\gamma \geq n|1 -1/2|$, $$ \begin{align*} \big\|\mathcal{H}^{-\beta /2}e^{it \mathcal{H}^{\gamma /2}}f\big\|_{L^p(\mathbb R^n)} \leq C (1 + |t|)^{n|1 -1/2|}\|f\|_{L^p(\mathbb R^n)}, \quad \forall t \in \mathbb{R}, \end{align*}$$and (b) for $p \in (0, 1]$, $\gamma& $ and $\beta /\gamma \geq n(1 -1/2)$, $$ \begin{align*} \big\|\mathcal{H}^{-\beta /2}e^{it \mathcal{H}^{\gamma /2}}f\big\|_{H^p_{\mathcal{H}}(\mathbb R^n)} \leq C (1 + |t|)^{n(1 -1/2)}\|f\|_{H^p_{\mathcal{H}}(\mathbb R^n)}, \quad \forall t \in \mathbb{R}, \end{align*}$$where $H^p_{\mathcal {H}}(\mathbb R^n)$ is the Hardy space associated with the operator $\mathcal {H}$. These estimates improve related result of Thangavelu [22] and have some interesting applications.
Publisher: Oxford University Press (OUP)
Date: 14-12-2021
DOI: 10.1093/IMRN/RNZ337
Abstract: Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for $L$ on the Besov and Triebel–Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$ but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.
Publisher: Mathematical Society of Japan (Project Euclid)
Date: 07-2012
Publisher: Walter de Gruyter GmbH
Date: 12-03-2015
Abstract: In this paper, we first introduce the new class of multiple weights A p → ∞ ${A^\\infty _{\\vec{p}}}$ which is larger than the class of multiple weights in [Adv. Math. 220 (2009), 1222–1264]. Then, using this class of weights, we study the weighted norm inequalities for certain classes of multilinear operators and their commutators with new BMO functions introduced in [J. Fourier Anal. Appl. 17 (2011), 115–134]. Finally, we show that some multilinear pseudodifferential operators fall within the scope of the theory obtained in this paper.
Publisher: Springer Science and Business Media LLC
Date: 22-02-2013
Publisher: Tokyo Journal of Mathematics
Date: 12-2014
Publisher: Walter de Gruyter GmbH
Date: 14-10-2018
Abstract: In this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.
Publisher: Springer Science and Business Media LLC
Date: 22-10-2016
Publisher: Elsevier BV
Date: 08-2008
Publisher: Elsevier BV
Date: 04-2020
Publisher: Wiley
Date: 25-07-2009
DOI: 10.1002/RNC.1383
Publisher: Springer Science and Business Media LLC
Date: 06-02-2009
Publisher: Elsevier BV
Date: 04-2021
Publisher: Informa UK Limited
Date: 18-11-2009
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2017
Publisher: American Mathematical Society (AMS)
Date: 05-07-2018
DOI: 10.1090/TRAN/7289
Abstract: Let X X be a space of homogeneous type and let L \\mathfrak {L} be a nonnegative self-adjoint operator on L 2 ( X ) L^2(X) enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove (local) nontangential and radial maximal function characterizations for the local Hardy spaces associated to L \\mathfrak {L} . This gives the maximal function characterization for local Hardy spaces in the sense of Coifman and Weiss provided that L \\mathfrak {L} satisfies certain extra conditions. Secondly we introduce local Hardy spaces associated with a critical function ρ \\rho which are motivated by the theory of Hardy spaces related to Schrödinger operators and of which include the local Hardy spaces of Coifman and Weiss as a special case. We then prove that these local Hardy spaces can be characterized by (local) nontangential and radial maximal functions related to L \\mathfrak {L} and ρ \\rho , and by global maximal functions associated to ‘perturbations’ of L \\mathfrak {L} . We apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Schrödinger operators on manifolds to Schrödinger operators on connected and simply connected nilpotent Lie groups.
Publisher: Indiana University Mathematics Journal
Date: 2020
Publisher: Hindawi Limited
Date: 2007
DOI: 10.1155/2007/26075
Abstract: We extend the classical Perron-Frobenius theorem for positive quasipolynomial matrices associated with homogeneous difference equations. Finally, the result obtained is applied to derive necessary and sufficient conditions for the stability of positive system.
Publisher: Springer Science and Business Media LLC
Date: 20-03-2017
Publisher: Elsevier BV
Date: 05-2020
Publisher: Springer Science and Business Media LLC
Date: 08-05-2009
Publisher: Springer Science and Business Media LLC
Date: 26-05-2017
Publisher: Elsevier BV
Date: 06-2020
Publisher: Cambridge University Press (CUP)
Date: 2020
DOI: 10.1017/FMS.2020.6
Abstract: Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper, we develop the theory of weighted Besov spaces ${\\dot{B}}_{p,q,w}^{\\unicode[STIX]{x1D6FC},L}(X)$ and weighted Triebel–Lizorkin spaces ${\\dot{F}}_{p,q,w}^{\\unicode[STIX]{x1D6FC},L}(X)$ associated with the operator $L$ for the full range $0 ,q\\leqslant \\infty$ , $\\unicode[STIX]{x1D6FC}\\in \\mathbb{R}$ and $w$ being in the Muckenhoupt weight class $A_{\\infty }$ . Under rather weak assumptions on $L$ as stated above, we prove that our new spaces satisfy important features such as continuous characterizations in terms of square functions, atomic decompositions and the identifications with some well-known function spaces such as Hardy-type spaces and Sobolev-type spaces. One of the highlights of our result is the characterization of these spaces via noncompactly supported functional calculus. An important by-product of this characterization is the characterization via the heat kernel for the full range of indices. Moreover, with extra assumptions on the operator $L$ , we prove that the new function spaces associated with $L$ coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of $L$ , the spectral multiplier of $L$ in our new function spaces and the dispersive estimates of wave equations.
Publisher: Springer Science and Business Media LLC
Date: 13-03-2014
Publisher: Elsevier BV
Date: 10-2008
Publisher: Springer Science and Business Media LLC
Date: 06-03-2020
Publisher: Elsevier BV
Date: 02-2017
Publisher: Springer Science and Business Media LLC
Date: 19-10-2013
Publisher: Elsevier BV
Date: 11-2010
Publisher: Elsevier BV
Date: 12-2021
Publisher: World Scientific Pub Co Pte Lt
Date: 30-08-2017
DOI: 10.1142/S0219199716500462
Abstract: In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions.
Publisher: Springer Science and Business Media LLC
Date: 02-12-2015
Publisher: Elsevier BV
Date: 05-2016
Publisher: Elsevier BV
Date: 02-2009
Publisher: Rocky Mountain Mathematics Consortium
Date: 10-2013
Publisher: Springer Science and Business Media LLC
Date: 12-05-2022
DOI: 10.1007/S00030-022-00765-4
Abstract: In this paper, we will study the Hardy and BMO spaces associated to the generalized Hardy operator $$L_{\alpha }= (-\Delta )^{\alpha /2}+a|x|^{-\alpha }$$ L α = ( - Δ ) α / 2 + a | x | - α . Similarly to the classical Hardy and BMO spaces, we will prove that our new function spaces will enjoy some important results such as molecular decomposition and duality. As applications, we show the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm inequalities involving the generalized Hardy operator.
Start Date: 05-2022
End Date: 05-2025
Amount: $375,000.00
Funder: Australian Research Council
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