Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators

We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inverse-power law interactions, and hard spheres. The functional spaces of these coercivity estimates depend on the collision kernel of these operators. They cover the spectral gap estimates for the linearized Boltzmann operator with Maxwell molecules, improve these estimates for hard potentials, and are the first explicit coercivity estimates for soft potentials (including in particular the case of Coulombian interactions). We also prove a regularity property for the linearized Boltzmann operator with non locally integrable collision kernels, and we deduce from it a new proof of the compactness of its resolvent for hard potentials without angular cutoff.     

Gaussian kernel operators on white noise functional spaces

The Gaussian kernel operators on white noise functional spaces, including second quantization, Fourier-Mehler transform, scaling, renormalization, etc. are studied by means of symbol calculus, and characterized by the intertwining relations with annihilation and creation operators. The infinitesimal generators of the Gaussian kernel operators are second order white noise operators of which the number operator and the Gross Laplacian are particular examples.     

Integral operators with the generalized sine kernel on the real axis

We study the asymptotic properties of integral operators with the generalized sine kernel acting on the real axis. We obtain the formulas for the Fredholm determinant and the resolvent in the large-x limit and consider some applications of the obtained results to the theory of integrable models.     

On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Navier–Stokes equations

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the L_q framework (1 < q < ∞), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularized traces of discrete operators

Unbounded perturbations of discrete operators are considered. Formulas for regularized traces are obtained, in which a finite number of corrections of the perturbation theory are used. An exact relation is established between the degree of subordination of a perturbation to the unperturbed operator and the number of corrections necessary for the existence of a finite formula of the trace. New estimates for the kernel norm of a resolvent of discrete operators are obtained.     

Boundedness of Marcinkiewicz integral on Triebel-Lizorkin spaces

In this paper, we prove the Triebel-Lizorkin boundedness for the Marcinkiewicz integral with rough kernel. The method we apply here enables us to consider more general operators.     

A General Sampling Theorem Associated with Differential Operators

In this paper we prove a general sampling theorem associated with differential operators with compact resolvent. Thus, we are able to recover, through a Lagrange-type interpolatory series, functions defined by means of a linear integral transform. The kernel of this transform is related with the resolvent of the differential operator. Most of the well-known sampling theorems associated with differential operators are shown to be nothing but limit cases of this result.     

Carnot Geometry and the Resolvent of the Sub-Laplacian for the Heisenberg Group

Abstract

We obtain an explicit representation formula for the sub-Laplacian on the isotropic, three-dimensional Heisenberg group. Using the formula we obtain themeromorphic continuation of the resolvent to the logarithmic plane, the existence of boundary values in the continuous spectrum, and semiclassical asymptotics of the resolvent kernel. The asymptotic formulas show the contribution of each Hamiltonian path in Carnot geometry to the spatial and high-energy asymptotics of the resolvent (convolution) kernel for the sub-Laplacian.     

Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups*

The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three-dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals–Gaveau–Greiner. Third, we prove that the uniform asymptotic behaviour at infinity (so the small-time asymptotic behaviour) of the heat kernel for Grushin operators, obtained by the first author, are still valid in two and three dimensions.     

The Hille-Yosida theorem for resolvent operators of multiparameter semigroups

We consider multiparameter semigroups of two types (multiplicative and coordinatewise) and resolvent operators associated with such semigroups. We prove an alternative version of the Hille-Yosida theorem in terms of resolvent operators. For simplicity of presentation, we give statements and proofs for two-parameter semigroups.     

Resolvent metric and the heat kernel estimate for random walks

In this paper we introduce the resolvent metric, the generalization of the resistance metric used for strongly recurrent walks. By using the properties of the resolvent metric we show heat kernel estimates for recurrent and transient random walks.     

Semiclassical resolvent estimates for two and three–body schrödinger operators

We prove uniform resolvent estimates for semiclassical three–body Schrödinger operators under a non–trapping condition for the classical flow of all subsystems. We also prove resolvent estimates for two–body Schrödinger operators with positive potentials when the energy level and the Planck constant tend both to zero.     

On the resolvent of singular Sturm-Liouville operators with transmission conditions

In this article, we investigate the resolvent operator of singular Sturm-Liouville problem with transmission conditions. We obtain integral representations for the resolvent of this operator in terms of the spectral function. Later, we discuss some properties of the resolvent operator, such as Hilbert-Schmidt kernel property, compactness. Finally, we give a formula in terms of the spectral function for the Weyl-Titchmarsh function of this problem.     

Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces H p σ and B p σ

In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel-Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 4, pp. 2–23, 2008 Original Russian Text Copyright ? by M. S. Agranovich To dear Israel Moiseevich Gelfand in connection with his 95th birthday Supported by RFBR grant no. 07-01-00287.     

Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l ²-norm for some general Mercer kernel matrices are provided. As an example, we give the l ²-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l ²-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms. Supported by the national NSF (No: 10871226) of P.R. China.     

Potentials Kernels Associated with a Filtration and Forward- Backward SDEs

In this paper, we study some operators associated with a filtration. We prove that these operators are Markov potential kernels closing a sub-Markov resolvent, using the method of continuation inspired by the study of forward-backward stochastic differential equations.     

The semiclassical resolvent on conformally compact manifolds with variable curvature at infinity

We construct a semiclassical parametrix for the resolvent of the Laplacian acting on functions on nontrapping conformally compact manifolds with variable sectional curvature at infinity. We apply this parametrix to analyze the Schwartz kernel of the semiclassical resolvent and Poisson operator and to show that the semiclassical scattering matrix is a semiclassical Fourier Integral Operator of appropriate class that quantizes the scattering relation. We also obtain high energy estimates for the resolvent and show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to obtain semiclassical resolvent estimates on certain conformally compact manifolds with hyperbolic trapping.     

Perturbation des Resolvantes Triangulaires et Balayage des Couples Excessifs

In this work, we start by developing an elementary potential theory associated to a triangular kernel. Then we study the perturbation of triangular resolvent, which enable us to characterize, with an elementary way, the excessive couples relatively to a triangular resolvent U. Finally, we study the balayage of U-surmedians couples and the triangular resolvents in duality.     

Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes

In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation ■,where ■ is the truncated fractional Laplacian,α∈(1,2) and b ∈ K_d~(α-1).In the second part,for a more general finite range jump process,we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x-y|in short time.     

Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT

We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.