Results 1 
9 of
9
Brownian motion and harmonic analysis on Sierpinski carpets
 MR MR1701339 (2000i:60083
, 1999
"... Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to con ..."
Abstract

Cited by 46 (9 self)
 Add to MetaCart
Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting. 1
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Random walks on graphical Sierpinski carpets
"... We consider random walks on a class of graphs derived from Sierpinski carpets. We obtain upper and lower bounds (which are nonGaussian) on the transition probabilities which are, up to constants, the best possible. We also extend some classical Sobolev and Poincare inequalities to this setting. ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
We consider random walks on a class of graphs derived from Sierpinski carpets. We obtain upper and lower bounds (which are nonGaussian) on the transition probabilities which are, up to constants, the best possible. We also extend some classical Sobolev and Poincare inequalities to this setting.
Várilly, “Dixmier traces on noncompact isospectral deformations
 J. Funct. Anal
"... We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of fu ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of R l, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.
Scattering at Obstacles of Finite Capacity
, 1996
"... For very general generators of diffusion semigroups we show that the essential and absolutely continuous spectra do not change when one adds an extra Dirichlet boundary condition on a "small" set. This is done by proving that the corresponding semigroup differences are HilbertSchmidt or trace class ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
For very general generators of diffusion semigroups we show that the essential and absolutely continuous spectra do not change when one adds an extra Dirichlet boundary condition on a "small" set. This is done by proving that the corresponding semigroup differences are HilbertSchmidt or trace class, respectively. Our method consists in a factorization argument which is based on calculating the semigroup difference via the FeynmanKac formula. We also derive trace class estimates for differences of resolvent powers, provided the underlying semigroup has finite local dimension.
Sobolev inequalities in familiar and unfamiliar settings
 In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1
The dual ultracontractivity and its applications
"... Let {Tt} be a symmetric Markov process on a measure space (M,m). The semigroup {Tt} is called ultracontractive if Tt are bounded operators from L 1 to L ∞ for all t>0. Some criteria for ultracontractivity are known. For example, for μ>0, the followings are equivalent to each other: ..."
Abstract
 Add to MetaCart
Let {Tt} be a symmetric Markov process on a measure space (M,m). The semigroup {Tt} is called ultracontractive if Tt are bounded operators from L 1 to L ∞ for all t>0. Some criteria for ultracontractivity are known. For example, for μ>0, the followings are equivalent to each other:
unknown title
, 2009
"... Isoperimetric profile of subgroups and probability of return of random walks on elementary solvable groups ∗ ..."
Abstract
 Add to MetaCart
Isoperimetric profile of subgroups and probability of return of random walks on elementary solvable groups ∗
Riesz transforms associated to Schrödinger operators with negative potentials
, 2009
"... The goal of this paper is to study the Riesz transforms ∇A−1/2 where A is the Schrödinger operator − ∆ − V, V ≥ 0, under different conditions on the potential V. We prove that if V is strongly subcritical, ∇A−1/2 is bounded on Lp (RN) , N ≥ 3, for all p ∈ (p ′ 0;2] where p ′ 0 is the dual exponent ..."
Abstract
 Add to MetaCart
The goal of this paper is to study the Riesz transforms ∇A−1/2 where A is the Schrödinger operator − ∆ − V, V ≥ 0, under different conditions on the potential V. We prove that if V is strongly subcritical, ∇A−1/2 is bounded on Lp (RN) , N ≥ 3, for all p ∈ (p ′ 0;2] where p ′ 0 is the dual exponent of p0 where 2 < 2N N−2 < p0 < ∞; and we give a counterexample to the boundedness on Lp (RN) for p ∈ (1;p ′ 0) ∪ (p0∗; ∞) where p0 ∗: = p0N is the reverse Sobolev expoN+p0 nent of p0. If the potential is strongly subcritical in the Kato subclass K ∞ N, then ∇A−1/2 is bounded on L p (R N) for all p ∈ (1;2], moreover if it is in L N/2 (R N) then ∇A −1/2 is bounded on L p (R N) for all p ∈ (1;N). We prove also boundedness of V 1/2 A −1/2 with the same conditions on the same spaces. Finally we study these operators on manifolds. We prove that our results hold on a class of Riemannian manifolds. keywords:Riesz transforms, Schrödinger operators, offdiagonal estimates, singular operators, Riemannian manifolds. Mathematics Subject Classification (2010): 42B20. 35J10. 1 Introduction and