The local part and the strong type for operators related to the Gaussian measure

In this work we present several theorems which imply the weak type 1 with respect to the Gaussian measure for the so-called local part of certain operators associated with the Ornstein-Uhlenbeck semigroup. Particular cases of these operators are the Riesz transforms of any order and the Littlewood-Paley square function. Also, we study general results based on the “size” of the operator which ensure the strong type 1 <p < ∞of both the local and global parts.     

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”.     

Invariant measures and regularity properties of perturbed Ornstein-Uhlenbeck semigroups

This paper deals with perturbations of the Ornstein-Uhlenbeck operator on L²-spaces with respect to a Gaussian measure μ. We perturb the generator of the Ornstein-Uhlenbeck semigroup by a certain unbounded, non-linear drift, and show various properties of the perturbed semigroup such as compactness and positivity. Strong Feller property, existence and uniqueness of an invariant measure are discussed as well.     

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”.     

Symmetric Ornstein-Uhlenbeck semigroups and their generators

 We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure μ. For a reversible process the domain of its generator in L ^p (μ) is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup (R _t ) to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of R _t and provide an estimate for the (appropriately defined) gradient of R _t φ. Finally, in the Hilbert-Schmidt case, we show tha t for any the function R _t φ is an (almost) classical solution of a version of the Kolmogorov equation. Received: 17 September 2001 / Revised version: 3 June 2002 / Published online: 30 September 2002 This work was partially supported by the Small ARC Grant Scheme. Mathematics Subject Classification (2000): Primary: 60H15, 47F05; Secondary: 60J60, 35R15, 35K15 Key words or phrases: Ornstein-Uhlenbeck operator – Second quantization – Reversibility – Spectral gap – Sobolev spaces – Domain of generator     

Semigroups and resolvents of bounded variation,imaginary powers andH ∞ functional calculus

Let −A be a linear, injective operator, on a Banach spaceX. We show that ∃ anH ^∞ functional calculus forA if and only if −A generates a bouned strongly continuous holomorphic semigroup of uniform weak bounded variation, if and only ifA(ζ+A) ⁻¹ is of uniform weak bounded variation. This provides a sufficient condition for the imaginary powers ofA, {A^−is} sεR, to extend to a strongly continuous group of bounded operators; we also give similar necessary conditions.     

Weyl Transforms,the Heat Kernel and Green Function of a Degenerate Elliptic Operator

We give a formula for the heat kernel of a degenerate elliptic partial differential operator L on ℝ² related to the Heisenberg group. The formula is derived by means of pseudo-differential operators of the Weyl type, {i.e.}, Weyl transforms, and the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for L²(ℝ²). Using the heat kernel, we give a formula for the Green function of L. Applications to the global hypoellipticity of L in the sense of tempered distributions, the ultracontractivity and hypercontractivity of the strongly continuous one-parameter semigroup e^−tL, t > 0, are given. Communicated by B.-W. Schulze (Potsdam) Mathematics Subject Classifications (2000): 47G30, 47E05.     

The Vector-Valued Big <Emphasis Type="Italic">q</Emphasis>-Jacobi Transform

Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L ²-space of functions on ℝ with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform ℱ with two different big q-Jacobi functions as a kernel, and we construct the inverse of ℱ.      

Weighted Inequalities for Multilinear Potential Operators and their Commutators

We prove weighted strong inequalities for the multilinear potential operator T_f{\cal T}_{\phi} and its commutator, where the kernel ϕ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. Moreover we prove weighted weak type inequalities for the multilinear maximal operator M_j,L^B\mathcal{M}_{\varphi,L^{B}} associated to an essentially nondecreasing function φ and to the Orlicz space L ^B for a given Young function B. This result allows us to obtain a weighted weak type inequality for the operator T_f{\cal T}_{\phi}.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), 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 prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L ²(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e ^?tL whose kernel p _t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L ^p -norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L ^p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L ² estimates of the kernels of the spectral multipliers.     

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.     

m-Dissipativity for Kolmogorov Operator of a Fractional Burgers Equation with Space-time White Noise

This paper is concerned with the essential m-dissipativity of the Kolmogorov operator associated with a fractional stochastic Burgers equation with space-time white noise. Some estimates on the solution and its moments with respect to the invariant measure are given. Moreover we also study the smoothing properties of the transition semigroup and the corresponding fractional Ornstein-Uhlenbeck operator by introducing an auxiliary semigroup and (generalized) Bismut-Elworthy formula. From these results, we prove that the Kolmogorov operator of the problem is m-dissipative and the domain of the infinitesimal generator of the fractional Ornstein-Uhlenbeck operator is a core.     

Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator T _k :L ₂(ν)→L ₂(ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of T _k . An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenvalues and -functions. It is well known that Mercer’s theorem has found important applications in various branches of mathematics, including probability theory and statistics. In particular, for some applications in the latter areas, however, it would be highly convenient to have a form of Mercer’s theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer’s theorem in this direction either stick too closely to the original topological structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical applications. In this work, we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in, e.g., statistical learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers of T _k in terms of interpolation spaces and investigate under which conditions these interpolation spaces are contained in L _∞(ν). For these two results, we then discuss applications related to the analysis of so-called least squares support vector machines, which are a state-of-the-art learning algorithm. Besides these results, we further use the obtained Mercer representations to show that every self-adjoint nuclear operator L ₂(ν)→L ₂(ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels.     

On the existence ofσ-finite invariant measures for operators

Several necessary and sufficient conditions are given for the existence of aσ-finite invariant measure for a positive operator onL _∞. They are ofσ-type: the entire space is an increasing union of setsX _k each of which is well-behaved. To the Memory of Shlomo Horowitz Research in part supported by the National Science Foundation (U.S.A.).     

The multi-dimensional pencil phenomenon for Laguerre heat-diffusion maximal operators

We investigate in detail the mapping properties of the maximal operator associated with the heat-diffusion semigroup corresponding to expansions with respect to multi-dimensional standard Laguerre functions . Our interest is focused on the situation when at least one coordinate of the type multi-index α is smaller than 0. For such parameters α the Laguerre semigroup does not satisfy the general theory of semigroups, and the behavior of the associated maximal operator on L ^p spaces is found to depend strongly on both α and the dimension. A. Nowak was supported in part by MNiSW Grant N201 054 32/4285.     

Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism

We study the existence of solutions to the orthogonal dynamics equation, which arises in the Mori-Zwanzig formalism in irreversible statistical mechanics. This equation generates the random noise associated with a reduction in the number of variables. IfL is the Liouvillian, or Lie derivative associated with a Hamiltonian system, andP an orthogonal projection onto a closed subspace ofL ², then the orthogonal dynamics is generated by the operator (I −P)L. We prove the existence of classical solutions for the case whereP has finite-dimensional range. In the general case, we prove the existence of weak solutions.     

Some spectral properties of the streaming operator with general boundary conditions

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C ₀-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case ‖K‖ ⩾ 1. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the solution of a characteristic equation.     

Ornstein�CUhlenbeck equations with time-dependent coefficients and L��vy noise in finite and infinite dimensions

We solve a time-dependent linear SPDE with additive Lévy noise in the mild and weak sense. Existence of a generalized invariant measure for the associated transition semigroup is established and the generator is studied on the corresponding L ²-space. The square field operator is characterized, allowing to derive a Poincaré and a Harnack inequality.     

Lois de sortie et semi-groupes basiques

Let ℙ=(P ^t ) ^t<0 be a semigroup of kernel and letm be an excessive reference measure for ℙ. In this work we prove that ℙ ism-basic if and only if everym.a.e. finite purely excessive function is represented by a unique exit law for ℙ. In this case we deduce some applications about natural densities, energie functionnal and invariant functions for the time-space semigroup of ℙ.