Results 1 
5 of
5
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
Central Gaussian semigroups of measures with continuous density
 J. Funct. Anal
, 1999
"... This paper investigates the existence and properties of symmetric central Gaussian semigroups ( t ) t>0 which are absolutely continuous and have a continuous density x 7! t (x), t > 0, with respect to Haar measure on groups of the form R n K where K is compact connected locally connected and ..."
Abstract

Cited by 8 (8 self)
 Add to MetaCart
This paper investigates the existence and properties of symmetric central Gaussian semigroups ( t ) t>0 which are absolutely continuous and have a continuous density x 7! t (x), t > 0, with respect to Haar measure on groups of the form R n K where K is compact connected locally connected and has a countable basis for its topology. We prove that there always exists a wealth of such Gaussian semigroups on any such group. For instance, if is any positive function increasing to innity, there exists a symmetric central Gaussian semigroup having a continuous density such that log t (e) log(1 + 1=t) (1=t) as t tends to zero. Among other results of this type we give a necessary and sucient condition on the structure of K for the existence of symmetric central Gaussian semigroups having a continuous density and such that t log t (e) is bounded above and below by positive constants for t 2 (0; 1) and some xed > 0. This condition is independent of . These results are proved by splitting any Gaussian semigroup (in a canonical way) into a semisimple part living on the commutator group G 0 and an Abelian part living on A = G=G 0 . For symmetric central Gaussian semigroups, we show that many properties hold for ( t ) t>0 if and only if they hold for both the semisimple and Abelian parts. This splitting principle is one of the main new tool developed of this paper. It leads to a much better understanding of central Gaussian semigroups and related objects. For instance, to any Gaussian convolution semigroup are associated a harmonic sheaf H and a quasidistance d on G. For symmetric central Gaussian semigroups on G = R n K, we show that H is a Brelot sheaf if and only if lim t!0 t log t (e) = 0. A sucient (but not necessary) condition...
On the hypoellipticity of subLaplacians on infinite dimensional compact groups.
, 2001
"... this paper we need the following denition. Denition 1.2 Let ( t ) t>0 be a convolution semigroup of measures. 1. We say that ( t ) t>0 has property (AC) if t is asbolutely continuous with respect to Haar measure, for all t > 0. 2. We say that ( t ) t>0 has property (CK) if it satises (AC) and has ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
this paper we need the following denition. Denition 1.2 Let ( t ) t>0 be a convolution semigroup of measures. 1. We say that ( t ) t>0 has property (AC) if t is asbolutely continuous with respect to Haar measure, for all t > 0. 2. We say that ( t ) t>0 has property (CK) if it satises (AC) and has a continuous density x 7! t (x) such that lim t!0 t log t (e) = 0: 2 Theorem 1.3 Let G be a compact connected group. Let L be the innitesimal generator of a symmetric Gaussian semigroup ( t ) t>0 . Let S be any topological space of continuous functions whose topology is not weaker than the uniform topology. 1. If G is not a Lie group, L is never B 0 (G)Shypoelliptic. 2. The operator L is L 1 (G)C(G)hypoelliptic if and only if ( t ) t>0 satises (AC). 3. Fix 1 p < +1 and assume that L is biinvariant. If L is L p (G)Shypoelliptic then ( t ) t>0 must satisfy (CK). Hypoellipticity properties are closely related to whether or not any harmonic distribution must be continuous. Denition 1.4 Let P be a leftinvariant dierential operator on G of nite order. Let A be a xed space of distributions. We say that P is Aregular if, for any domain , each U 2 B 0 (G) such that 8 2 B 0( ; U 2 A and Z PUd = 0; can be represented in by a continuous function. We will prove a version of Theorem 1.3 where hypoellipticity properties are replaced by the corresponding regularity conditions in the sense of Denition 1.4. Note that this denition requires the distribution U to have two dierent properties in (a) 8 2 B 0( 7 U 2 A, which is a regularity property; (b) U is a solution of PU = 0 in Condition (a) plays a crucial role in the present context. In the companion paper [8], we prove some strong hypoellipticity results under the h...
Nash Type Inequalities for Fractional Powers of NonNegative SelfAdjoint Operators
"... Assuming that a Nash type inequality is satisfied by a nonnegative selfadjoint operator A, we prove a Nash type inequality for the fractional powers of A. Under some assumptions, we give ultracontractivity bounds for the semigroup (T t; ) generated by A . ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Assuming that a Nash type inequality is satisfied by a nonnegative selfadjoint operator A, we prove a Nash type inequality for the fractional powers of A. Under some assumptions, we give ultracontractivity bounds for the semigroup (T t; ) generated by A .
Gaussian Bounds For Derivatives Of Central Gaussian Semigroups On Compact Groups
"... . For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumption, we prove that these derivatives satisfy certain Gaussian ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
. For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumption, we prove that these derivatives satisfy certain Gaussian bounds. 1.