Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

The Ornstein-Uhlenbeck semigroup in exterior domains

Let Ω be an exterior domain in It is shown that Ornstein-Uhlenbeck operators L generate C₀-semigroups on L^p(Ω) for p ∈ (1, ∞) provided ∂Ω is smooth. The method presented also allows to determine the domain D(L) of L and to prove L^p − L^q smoothing properties of e^tL. If ∂Ω is only Lipschitz, results of this type are shown to be true for p close to 2. Received: 16 December 2004; revised: 4 February 2005     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .     

Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations

We study a certain class of von Neumann algebras generated by selfadjoint elements ω_i=a_i+a_i⁺, where a_i, a_i⁺ satisfy the general commutation relations:We assume that operator T for which the constants are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality were shown. In this paper we prove that T-Ornstein-Uhlenbeck semigroup U_t=Γ_T(e^−t), t>0 arising from the second quantization procedure is hyper- and ultracontractive. The optimal bounds for hypercontractivity are also discussed.This paper was partially supported by KBN grant no 2P03A00732 and also by RTN grant HPRN-CT-2002-00279.     

Pointwise and norm estimates for operators associated with the Ornstein-Uhlenbeck semigroup

In this Note, we find the explicit expression for the pointwise supremum of the kernels defining the Ornstein-Uhlenbeck semigroup. This allows us to give a more geometric proof of the weak type 1 for the Maximal Mehler transform, given in [7]. In addition, direct proofs of several results about the boundedness of the Riesz transforms of any order and the Littlewood-Paley g function associated to this problem are presented.     

Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup

We find necessary and sufficient conditions on a Banach spaceX in order for the vector-valued extensions of several operators associated to the Ornstein-Uhlenbeck semigroup to be of weak type (1, 1) or strong type (p, p) in the range 1<p<∞. In this setting, we consider the Riesz transforms and the Littlewood-Paleyg-functions. We also deal with vector-valued extensions of some maximal operators like the maximal operators of the Ornstein-Uhlenbeck and the corresponding Poisson semigroups and the maximal function with respect to the gaussian measure. In all cases, we show that the condition onX is the same as that required for the corresponding harmonic operator: UMD, Lusin cotype 2 and Hardy-Littlewood property. In doing so, we also find some new equivalences even for the harmonic case. The first and third authors were partially supported by CONICET (Argentina) and Convenio Universidad Autónoma de Madrid-Universidad Nacional del Litoral. The second author was partially supported by the European Commission via the TMR network “Harmonic Analysis”.     

Ornstein-Uhlenbeck半群证明不等式的一些应用

按照Ornstein-Uhlenbeck的思想方法,用Ornstein-Uhlenbeck半群和Ornstein-Uhlenbeck算子的一些重要性质,对Brascamp-Lieb不等式、高斯对数Sobolev不等式、逆Bobkov等周不等式等几个重要的几何与分析不等式给出了另一证明.     

On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup

In this paper, for each given $ we characterize the weights v for which the centered maximal function with respect to the gaussian measure and the Ornstein-Uhlenbeck maximal operator are well defined for every function in and their means converge almost everywhere. In doing so, we find that this condition is also necessary and sufficient for the existence of a weight u such that the operators are bounded from into We approach the poblem by proving some vector valued inequalities. As a byproduct we obtain the strong type (1,1) for the “global” part of the centered maximal function. Received May 18, 1999 / Revised December 9, 1999 Published online July 20, 2000     

On the Dirichlet semigroup for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

We consider a family of self-adjoint Ornstein-Uhlenbeck operators Lα in an infinite dimensional Hilbert space H having the same gaussian invariant measure μ for all α∈[0,1]. We study the Dirichlet problem for the equation λφ−Lαφ=f in a closed set K, with f∈L²(K,μ). We first prove that the variational solution, trivially provided by the Lax-Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to K. Then we address two problems: 1) the regularity of the solution φ (which is by definition in a Sobolev space ) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside K belongs to . In the second case we exploit the Malliavin's theory of surface integrals which is recalled in Appendix A of the paper, then we are able to give a meaning to the trace of φ at ∂K and to show that it vanishes, as it is natural.     

Maximal operators related to the Ornstein-Uhlenbeck semigroup with complex time parameter

Let γ be the Gaussian measure in Rd and Ht, t>0, the corresponding Ornstein-Uhlenbeck semigroup, whose infinitesimal generator is . For each p with 1<p<∞, let Ep⊂C be the closure of the region of holomorphy of the map t?Ht taking values in the space of bounded operators on Lp(γ). We examine the maximal operator . The known results about concern mainly the case p<2. We prove that for p>2 this operator is of weak type but not of strong type (p,p) for γ. However, if a neighbourhood of the origin is deleted from Ep in the definition of , the resulting operator is shown to be of strong type.     

Spectral decompositions and decomposable multipliers

A perturbation result for spectral decompositions of several commuting operators is developed. Roughly speaking, we prove that a composed system (S,T) consisting of commuting continuous linear Banach space operators admits spectral decompositions, if the restrictions of S onto and the quotients of S modulo the spectral subspaces of T admit spectral decompositions. In the second part it is shown that this result has a natural application to systems of multipliers on complex Banach algebras with bounded approximate identity.Dedicated to my teacher Professor H. G. Tillmann on his 60th birthday     

Spectral Properties of Uperpositions of Ornstein-Uhlenbeck Type Processes

Stationary processes with prescribed one-dimensional marginal laws and long-range dependence are constructed. The asymptotic properties of the spectral densities are studied. The possibility of Mittag-Leffler decay in the autocorrelation function of superpositions of Ornstein-Uhlenbeck type processes is proved. AMS 2000 Subject Classification Primary 60E07, 60G10, 60G18 Secondary 62M10, 62P05 Supported by the Danish National Research Foundation and EPSRC grant RCMT091. Partially supported by the ARC grants DP 0345577, DP0559807 and EPSRC grant RCMT091.     

Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure , whose density with respect to the left Haar measure λ_G is a nontrivial positive character of G. We show that if p≠2 and G is amenable, then every spectral multiplier of extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are spectral multipliers of . Work partially supported by the EC HARP Network “Harmonic Analysis and Related Problems”, the Progetto Cofinanziato MURST “Analisi Armonica” and the Gruppo Nazionale INdAM per l'Analisi Matematica, la Probabilità e le loro Applicazioni. Part of this work was done while the second and the third author were visiting the “Centro De Giorgi” at the Scuola Normale Superiore di Pisa, during a special trimester in Harmonic Analysis. They would like to express their gratitude to the Centro for the hospitality.     

Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups

Let G = N ? A, where N is a stratified group and A = ? acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G, and their sum is a sub-Laplacian Δ on G. We prove a theorem of Mihlin–Hörmander type for spectral multipliers of Δ. The proof of the theorem hinges on a Calderón–Zygmund theory adapted to a sub-Riemannian structure of G and on L¹-estimates of the gradient of the heat kernel associated to the sub-Laplacian Δ.     

Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup

Let (S,ω) be a weighted abelian semigroup, let M _ω (S) be the semigroup of ω-bounded multipliers of S, and let \(\mathcal {A}\) be a strictly convex commutative Banach algebra with identity. It is shown that T is an onto isometric multiplier of \(\ell ^{1}(S,\omega , \mathcal {A})\) if and only if there exists an invertible σ ∈ M _ω (S), a unitary point \(a \in \mathcal {A}\), and a k>0 such that \(T(f)= ka{\sum }_{x \in S} f(x)\delta _{\sigma (x)}\) for each \(f={\sum }_{x \in S}f(x)\delta _{x} \in \ell ^{1}(S,\omega ,\mathcal {A})\). It is also shown that an isomorphism from \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\) onto \(\ell ^{1}(S_{2},\omega _{2}, \mathcal {B})\) induces an isomorphism from \(M(\ell ^{1}(S_{1},\omega _{1},\mathcal {A}))\), the set of all multipliers of \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\), onto \(M(\ell ^{1}(S_{2},\omega _{2},\mathcal {B}))\).     

The Ornstein-Uhlenbeck semigroup in an infinite dimensional L2 setting

We study the number operator, N, of quantum field theory as a partial differential operator in infinitely many variables. Informally Nu(x) = ?Δu(x) + x · grad u(x). A large core for N is constructed which is invariant under e^?tN and on which this informal expression may be given a precise and natural meaning.