Results 1  10
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19
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 52 (11 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
Riesz transform on manifolds and Poincaré inequalities
, 2005
"... We study the validity of the L p inequality for the Riesz transform when p> 2 and of its reverse inequality when p < 2 on complete Riemannian manifolds under the doubling property and some Poincaré inequalities. ..."
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Cited by 17 (7 self)
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We study the validity of the L p inequality for the Riesz transform when p> 2 and of its reverse inequality when p < 2 on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials
, 2006
"... We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved FeffermanPhong inequalities and reverse Hölder estimates for weak solutions of − ∆ + ..."
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Cited by 8 (3 self)
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We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved FeffermanPhong inequalities and reverse Hölder estimates for weak solutions of − ∆ + V and their gradients.
Some constructions for the fractional Laplacian on noncompact manifolds
"... We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by CaffarelliSilvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at ..."
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Cited by 6 (0 self)
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We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by CaffarelliSilvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at infinity and geometry plays a crucial role. First we give explicit calculations in the hyperbolic space, including a formula for the kernel and a trace Sobolev inequality. Then we consider more general noncompact manifolds, where the problem reduces to obtain suitable upper bounds for the heat kernel. Contents 1 Introduction and statement of the results 2
Harmonic analysis related to Schrödinger operators, arXiv:0711.3262v1. 24 W. Schlag, A remark on LittlewoodPaley theory for the distorted Fourier transform
 Proc. Amer. Math. Soc
, 2007
"... Abstract. In this article we give an overview on some recent development of LittlewoodPaley theory for Schrödinger operators. We extend the LittlewoodPaley theory for special potentials considered in the authors ’ previous work. We elaborate our approach by considering potential in C ∞ 0 or Schwar ..."
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Cited by 2 (1 self)
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Abstract. In this article we give an overview on some recent development of LittlewoodPaley theory for Schrödinger operators. We extend the LittlewoodPaley theory for special potentials considered in the authors ’ previous work. We elaborate our approach by considering potential in C ∞ 0 or Schwartz class in one dimension. In particular the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give maximal function characterization of the Besov spaces and TriebelLizorkin spaces associated with H. We then prove a spectral multiplier theorem on these spaces and derive Strichartz estimates for the wave equation with a potential. We also consider similar problem for the unbounded potentials in the Hermite and Laguerre cases, whose V = ax  2 + bx  −2 are known to be critical in the study of perturbation of nonlinear dispersive equations. This improves upon the previous results when we apply the upper Gaussian bound for the heat kernel and its gradient. 1.
RIESZ TRANSFORM ON MANIFOLDS WITH QUADRATIC CURVATURE DECAY.
"... ABSTRACT. We investigate the Lpboundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the Lpunboundness of the Riesz transform are given. We recover known results about manifolds that are Euclidean or conical at infinity. RÉSUME ́ : On ..."
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Cited by 1 (0 self)
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ABSTRACT. We investigate the Lpboundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the Lpunboundness of the Riesz transform are given. We recover known results about manifolds that are Euclidean or conical at infinity. RÉSUME ́ : On étudie la continuite ́ de la transformée de Riesz sur les espaces Lp pour des variétés dont la courbure de Ricci décroit quadratiquement. Nous donnons aussi deux critères géométriques impliquant la non continuite ́ de la transformée de Riesz. Notre méthode nous permet de retrouver les résultats connus pour les variétés euclidiennes ou coniques a ̀ l’infini.
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"... quadrature formula for diffusion polynomials corresponding to a generalized heat kernel ..."
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quadrature formula for diffusion polynomials corresponding to a generalized heat kernel
Riesz transform on manifolds and heat . . .
, 2004
"... One considers the class of complete noncompact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel s ..."
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One considers the class of complete noncompact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same