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.     

Weighted Norm Inequalities for Spectral Multipliers on Graphs

Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by \(Pf(x) = {\sum }_{y} p(x, y)f(y)\). We assume that the kernels p_n(x, y) associated to Pⁿ satisfy Gaussian upper bounds but do not assume they satisfy the Hölder continuity property and the temporal regularity. Denote by L = I ? P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted L^p norm inequalities for the spectral multipliers of L. We also obtain the weighted L^p norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case.     

Plancherel-type estimates and sharp spectral multipliers

We study general spectral multiplier theorems for self-adjoint positive definite operators on L²(X,μ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R³ and new spectral multiplier theorems for the Laguerre and Hermite expansions.     

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Let L   be a non-negative self-adjoint operator acting on L²(X)$L^2(X)$ where X is a space of homogeneous type. Assume that L   generates a holomorphic semigroup e^−tL$e^{-tL}$ which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.     

Weighted Norm Inequalities for Commutators of BMO Functions and Singular Integral Operators with Non-Smooth Kernels

The aim of this paper is to establish a sufficient condition for certain weighted norm inequalities for singular integral operators with non-smooth kernels and for the commutators of these singular integrals with BMO functions. Our condition is applicable to various singular integral operators, such as the second derivatives of Green operators associated with Dirichlet and Neumann problems on convex domains, the spectral multipliers of non-negative self-adjoint operators with Gaussian upper bounds, and the Riesz transforms associated with magnetic Schrödinger operators.     

Pseudodifferential operators on Bochner spaces and an application

We consider pseudodifferential operators with operator valued symbols a(x,ξ) acting on a UMD Banach space X. Assuming some regularity of Hölder type in x and Mihlin type in ξ we prove L _p (? ⁿ ;X) boundedness of such operators. This result is then applied to the study of L _p -maximal regularity for nonautonomous parabolic evolution equations.     

Norm convergence of multiple ergodic averages on amenable groups

We consider abstract non-negative self-adjoint operators on L²(X) which satisfy the finite-speed propagation property for the corresponding wave equation. For such operators, we introduce a restriction type condition, which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next, we show that in the considered abstract setting, our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on Rⁿ, the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.     

Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type

Let X be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, via a new Cotlar type inequality linking commutators and corresponding maximal operators, a weighted Lp(X) estimate with general weights and a weak type endpoint estimate with A₁(X) weights are established for maximal operators corresponding to commutators of BMO(X) functions and singular integral operators with non-smooth kernels.     

Banach space sequences and projective tensor products

In this paper we use results from the theory of tensor products of Banach spaces to establish the isometry of the space of (1,p)-summing sequences (also known as strongly p-summable sequences) in a Banach space X, the space of nuclear X-valued operators on ?^p and the complete projective tensor product of ?^p with X. Through similar techniques from the theory of tensor products, the isometry of the sequence space L^p〈X〉 (recently introduced in a paper by Bu, Quaestiones Math. (2002), to appear), the space of nuclear X-valued operators on L^p(0,1) (with a suitable equivalent norm) and the complete projective tensor product of L^p(0,1) with X is established. Moreover, we find conditions for the space of (p,q)-summing multipliers to have the GAK-property (generalized AK-property), use multiplier sequences to characterize Banach space valued bounded linear operators on the vector sequence space of absolutely p-summable sequences in a Banach space and present short proofs for results on p-summing multipliers.     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

A variational approach to dislocation problems for periodic Schrödinger operators

As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential V=V(x,y) on R² with period lattice Z² by setting Wt(x,y)=V(x+t,y) for x<0 and Wt(x,y)=V(x,y) for x?0, for t∈[0,1]. For Lipschitz-continuous V it is shown that the Schrödinger operators Ht=−Δ+Wt have spectrum (surface states) in the spectral gaps of H₀, for suitable t∈(0,1). We also discuss the density of these surface states as compared to the density of the bulk. Our approach is variational and it is first applied to the well-known dislocation problem (Korotyaev (2000, 2005) [15] and [16]) on the real line. We then proceed to the dislocation problem for an infinite strip and for the plane. In Appendix A, we discuss regularity properties of the eigenvalue branches in the one-dimensional dislocation problem for suitable classes of potentials.     

Boundedness properties for a class of integral operators including the index transforms and the operators with complex Gaussian kernels

In this paper we study the behavior of general integral operators on weighted L^p spaces. Particular cases include the main index transforms and the operators with complex Gaussian kernels. We also extend some previous results established in [E.R. Negrin, Proc. Amer. Math. Soc. 123 (1995) 1185-1190].     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Heat kernel bounds and desingularizing weights

We study the parabolic operator ∂_t−Δ_x+V(t,x), in d?1, with a potential V=V+−V−,V±?0 assumed to be from a parabolic Kato class, and obtain two-sided Gaussian bounds on the associated heat kernel. The constraints on the Kato norms of V⁺ and V⁻ are completely asymmetric, as they should be. Further improvements to our heat kernel bounds are obtained in the case of time-independent potentials.If V has singularities of the type ±c|x|⁻² with a suitably small constant c, we obtain new lower and (sharp) upper weighted heat kernel bounds. The rate of growth of the weights depends (explicitly) on the constant c. The standard bounds and methods (estimates in L^p-spaces without desingularizing weights) fail for singular potentials.     

A generalised Christoffel-Darboux formula and reproducing kernels in spaces of polymonogenic and polyharmonic functions

Let ?_x be the Dirac operator in R^m+1, let t ∈ N and let K_t(x,y) be the Bergman kernel for the space L ₂M_t of square integrable polymonogenic functions of order t in the unit ball B(1) of R^m+1. Then expressions for K_t(x,y) are derived using a generalised Christoffel-Darboux formula related to the orthogonal basis of generalised Gegenbauer polynomials in the weighted L₂-space L₂(B(1);(1 + x²⁾α), α > ?1.     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

Comportement des noyaux de la chaleur des opérateurs de Schrödinger et applications à certaines équations paraboliques semi-linéaires

We consider general Schrödinger operators on domains of Riemannian manifolds with possibly exponential volume growth. We prove sharp large time Gaussian upper bounds. These bounds are then used to prove new Lp-Lp estimates for the corresponding semigroups. Applications to semi-linear parabolic equations are given.     

Properties of the density for a three-dimensional stochastic wave equation

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let pt,x(y) be the density of the law of the solution u(t,x) of such an equation at points (t,x)∈]0,T]×R³. We prove that the mapping (t,x)?pt,x(y) owns the same regularity as the sample paths of the process {u(t,x),(t,x)∈]0,T]×R³} established in [R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press]. The proof relies on Malliavin calculus and more explicitly, the integration by parts formula of [S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./Springer-Verlag, Bombay, 1984] and estimates derived from it.