Results 1 
9 of
9
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Real interpolation of Sobolev spaces
, 2008
"... We prove that W 1 p is an interpolation space between W 1 p1 and W 1 p2 for p> q0 and 1 ≤ p1 < p < p2 ≤ ∞ on some classes of manifolds and general metric spaces, where q0 depends on our hypotheses. ..."
Abstract

Cited by 6 (6 self)
 Add to MetaCart
We prove that W 1 p is an interpolation space between W 1 p1 and W 1 p2 for p> q0 and 1 ≤ p1 < p < p2 ≤ ∞ on some classes of manifolds and general metric spaces, where q0 depends on our hypotheses.
Harnack inequality and hyperbolicity for subelliptic pLaplacians with applications to Picard type theorems
, 2000
"... Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The pLaplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The nonsmooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 pparabolicity and phyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and pparabolicity . . . . . . . . . . . . . . . . .
REAL INTERPOLATION OF SOBOLEV SPACES ASSOCIATED TO A Weight
, 2008
"... We study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V, arising from Schrödinger operators with positive potential. We show that for 1 ≤ p1 < p < p2 < q0 with p> s0, W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of mani ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V, arising from Schrödinger operators with positive potential. We show that for 1 ≤ p1 < p < p2 < q0 with p> s0, W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of manifolds and Lie groups. The constants s0, q0 depend on our hypotheses.
L p Boundedness of Riesz transform related to Schrödinger operators on a manifold
 ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA, CLASSE DI SCIENZE (2009) 725765
, 2009
"... We establish various L p estimates for the Schrödinger operator − ∆ + V on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where ∆ is the LaplaceBeltrami operator and V belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups wit ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We establish various L p estimates for the Schrödinger operator − ∆ + V on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where ∆ is the LaplaceBeltrami operator and V belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial growth.
Non local poincaré inequalities on lie groups with polynomial volume growth. http://jp.arxiv.org/abs/1001.4075, 2010. Emmanuel Russ– Université Paul Cézanne, LATP, Faculté des Sciences et Techniques, Case cour A Avenue Escadrille NormandieNiemen
 F13397 Marseille, Cedex 20, France et CNRS, LATP, CMI, 39 rue F. JoliotCurie, F13453 Marseille Cedex 13, France HARDY INEQUALITIES 11 Yannick Sire– Université Paul Cézanne, LATP, Faculté des Sciences et Techniques, Case cour A Avenue Escadrille Normand
"... Abstract. Let G be a real connected Lie group with polynomial volume growth, endowed with its Haar measure dx. Given a C 2 positive function M on G, we give a sufficient condition for an L 2 Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaréin ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. Let G be a real connected Lie group with polynomial volume growth, endowed with its Haar measure dx. Given a C 2 positive function M on G, we give a sufficient condition for an L 2 Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaréinequality on G with respect to M(x)dx.
Subelliptic pharmonic maps into spheres and the ghost of Hardy spaces
 MATH. ANN. 312, 341–362
, 1998
"... ..."
Equivalence of Coupling and Shift Coupling
, 1999
"... It is proved in this paper that a weak parabolic Harnack inequality for a Markov semigroup implies the existence of a coupling and a shift coupling for the corresponding process with equal chances of success. This implies equality of the tail and invariant σfields for the diffusion as well as equa ..."
Abstract
 Add to MetaCart
It is proved in this paper that a weak parabolic Harnack inequality for a Markov semigroup implies the existence of a coupling and a shift coupling for the corresponding process with equal chances of success. This implies equality of the tail and invariant σfields for the diffusion as well as equality of the class of bounded parabolic functions and the class of bounded harmonic functions.