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Discovery Projects - Grant ID: DP130101302

Title
Discovery Projects - Grant ID: DP130101302
Funding
ARC | Discovery Projects
Contract (GA) number
DP130101302
Start Date
2013/01/01
End Date
2015/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP130101302

 

  • HARDY SPACES ON METRIC MEASURE SPACES WITH GENERALIZED SUB-GAUSSIAN HEAT KERNEL ESTIMATES

    Accepted for publication; International audience; Hardy space theory has been studied on manifolds or metric measure spaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However, there are natural examples where one finds a mix of both behaviour (locally Gaussian and at infinity sub-Gaussian) in which case the previous theory doesn't apply. Still we define molecular and square function Hardy spaces using appropriate scaling, and we show that they agree with Lebesgue spa...

    Probabilistic approach to quantum separation effect for Feynman-Kac semigroup

    Sikora, Adam; Zienkiewicz, Jacek (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP130101302 (DP130101302)
    Quantum tunnelling phenomenon allows a particle in Schr\"odinger mechanics tunnels through a barrier that it classically could not overcome. Even the infinite potentials do not always form impenetrable barriers. We discuss an answer to the following question: What is a critical magnitude of potential, which creates impenetrable barrier and for which the corresponding Schr\"odinger evolution system separates? In addition we describe some quantitative estimates for the separating effect in term...

    The limitations of the Poincar{\'e} inequality

    Robinson, Derek W.; Sikora, Adam (2013)
    Projects: ARC | Discovery Projects - Grant ID: DP130101302 (DP130101302)
    We examine the validity of the Poincar\'e inequality for degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where $H_\delta$ is a generalized Gru\v{s}in operator, \[ H_\delta=-\nabla_{x_1}\,|x_1|^{(2\delta_1,2\delta_1')}\,\nabla_{x_1}-|x_1|^{(2\delta_2,2\delta_2')}\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$, $\delta_1,\delta_...

    Sharp Heat Kernel Bounds and Entropy in Metric Measure Spaces

    We establish sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying $\RCD(0,N)$ ( equivalently, $\RCD^\ast(0,N)$) condition with $N\in \mathbb{N}\setminus\{1\}$ and having maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the (minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity ...

    Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces

    Wang, Yuzhao; Li, Huaiqian (2015)
    Projects: ARC | Discovery Projects - Grant ID: DP130101302 (DP130101302)
    New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--\'{E}mery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the assumption of nonnegative $\infty$-Bakry--\'{E}mery Ricci curvature and the $m$-Bakry--\'{E}mery Ricci curvature bounded from below by a non-positive constant, the Li--Yau ...

    Gradient estimates for heat kernels and harmonic functions

    Coulhon, Thierry; Jiang, Renjin; Koskela, Pekka; Sikora, Adam (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP130101302 (DP130101302)
    Let $(X,d,\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carr\'e du champ". Assume that $(X,d,\mu,\E)$ supports a scale-invariant $L^2$-Poincar\'e inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse H\"older inequality for the gradients of harmonic fun...
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