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.     

Maximal functions for multipliers on stratified groups

In this paper we obtain a refined $L^p$ bound for maximal functions of the multiplier operators on stratified groups and maximal functions of the multi‐parameter multipliers on product spaces of stratified groups. As an application we find a refined $L^p$ bound for maximal functions of joint spectral multipliers on Heisenberg group.     

Local spectral theory for normal operators in Krein spaces

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.     

Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group

Consider the group of affine transformations of the line. Denote by X and Y the right-invariant vector fields corresponding to the s and t directions, respectively, and let We prove that the first-order Riesz operator is of weak type (1, 1) with respect to left Haar measure. This operator is therefore also bounded on . Our results provide answers, in a particular instance, to the open question of the boundedness of Riesz operators on Lie groups of exponential growth. The main parts of the proof concern the behaviour of the kernel of the operator at infinity, and exploit cancellation. A key technique is to use expansion with respect to scales of Haar-like functions. Received March 16, 1998; in final form June 22, 1998     

Paley–Littlewood decomposition for sectorial operators and interpolation spaces

We prove Paley–Littlewood decompositions for the scales of fractional powers of 0‐sectorial operators A on a Banach space which correspond to Triebel–Lizorkin spaces and the scale of Besov spaces if A is the classical Laplace operator on We use the ‐calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace‐type operators on manifolds and graphs, Schrödinger operators and Hermite expansion. We also give variants of these results for bisectorial operators and for generators of groups with a bounded ‐calculus on strips.     

Compact multipliers on spaces of analytic functions

In the paper compact multiplier operators from Banach spaces of analytic functions on the unit disk into Banach sequence lattices are studied. If , then the characterization of compact multipliers is obtained through calculating the Hausdorff measure of noncompactness of diagonal operators between Banach sequence lattices. Furthermore, in the general case , necessary and sufficient conditions for compactness are presented. Received: 12 August 2008, Revised: 11 January 2009     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

Hypercyclic composition operators on Hv0‐spaces

We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator C_φ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators C_φ on $H_{v}^0$.     

Laplace’s method for iterated complex Brownian integrals

A kind of Laplace’s method is developped for iterated stochastic integrals where integrators are complex standard Brownian motions. Then it is used to extend properties of Bougerol and Jeulin’s path transform in the random case when simple representations of complex semisimple Lie algebras are not supposed to be minuscule.     

Spectral theory for the fractal Laplacian in the context of h‐sets

An h‐set is a nonempty compact subset of the Euclidean n‐space which supports a finite Radon measure for which the measure of balls centered on the subset is essentially given by the image of their radius by a suitable function h. In most cases of interest such a subset has Lebesgue measure zero and has a fractal structure. Let Ω be a bounded C^∞ domain in with Γ ? Ω. Let where (?Δ)^?1 is the inverse of the Dirichlet Laplacian in Ω and tr^Γ is, say, trace type operator. The operator B, acting in convenient function spaces in Ω, is studied. Estimations for the eigenvalues of B are presented, and generally shown to be dependent on h, and the smoothness of the associated eigenfunctions is discussed. Some results on Besov spaces of generalised smoothness on and on domains which were obtained in the course of this work are also presented, namely pointwise multipliers, the existence of a universal extension operator, interpolation with function parameter and mapping properties of the Dirichlet Laplacian. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on , namely We assume that and is a nonnegative function. By definition, a function belongs to the Hardy space if Under certain assumptions on V we characterize the space in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to for with no additional assumptions on the potential V.     

Function theory in real Hardy spaces

We show that many classical results in Hardy space theory have exact analogues when the Fourier coefficients are restricted to be real. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ? Kr^N, N ∈ (0, ∞). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space L^p(?)(X, μ) into a certain Musielak‐Orlicz space L^p(X, μ) with the N‐function Φ(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska‐Orlicz indices of the function a(r) is also given. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Gustafson,weidmann, kato,wolf, schechter and browder essential spectra of some matrix operator and application to two‐group transport equation

In this paper, we investigate the Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of a 2 × 2 block matrix operator defined on a Banach space where entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components under some properties. Furthermore, we give an application to two‐group transport equation. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Spectral approximation of banded Laurent matrices with localized random perturbations

This paper explores the relationship between the spectra of perturbed infinite banded Laurent matricesL(a)+K and their approximations by perturbed circulant matricesC _n (a)+P _n KP _n for largen. The entriesK _jk of the perturbation matrices assume values in prescribed sets _jk at the sites (j, k) of a fixed finite setE, and are zero at the sites (j, k) outsideE. WithK ^E denoting the ensemble of these perturbation matrices, it is shown that

Existence results for asymmetric fractional ‐Laplacian problem

We investigated a class of quasi‐linear nonlocal problems with a right‐hand side nonlinearity which exhibits an asymmetric growth at and . Namely, it is linear at and superlinear at . However, it needs not satisfy the Ambrosetti–Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by using variational methods combined with the Moser–Trudinger inequality.     

Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators H on with coefficients invariant under translation by can be analyzed through decomposition in terms of versions , , of H with z-periodic boundary conditions acting on where . If the s emigroup S generated by H has a H?lder continuous integral kernel satisfying Gaussian bounds then the semigroups generated by the have kernels with similar properties and extends to a function on which is analytic with respect to the trace norm. The sequence of semigroups obtained by rescaling the coefficients of by converges in trace norm to the semigroup generated by the homogenization of . These convergence properties allow asymptotic analysis of the spectrum of H. Received September 1, 1998; in final form January 14, 1999