On a Class of Integral Operators

In this paper we consider the space where dv _s is the Gaussian probability measure. We give necessary and sufficient conditions for the boundedness of some classes of integral operators on these spaces. These operators are generalizations of the classical Bergman projection operator induced by kernel function of Fock spaces over .      

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Fredholm conditions and an index formula are obtained for Wiener-Hopf operators W(a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w is a Muckenhoupt weight on and . The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on that are continuous on and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SO_{p, w} of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderón-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

Spectral Properties of Some Linear Matrix Differential Operators in Lp-spaces on $${\mathbb{R}}$$

A detailed study is made of matrix-valued, ordinary linear differential operators T in for 1 < p < ∞, which arise as the perturbation of a constant coefficient differential operator of order n ≥ 1 by a lower order differential operator which has a factorisation S = AB for suitable operators A and B. Via techniques from L ^p -harmonic analysis, perturbation theory and local spectral theory, it is shown that T satisfies certain local resolvent estimates, which imply the existence of local functional calculi and decomposability properties of T.      

C*-Algebras of Singular Integral Operators with Shifts Having the Same Nonempty Set of Fixed Points

The C*-subalgebra of generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms with any nonempty set of common fixed points is studied. A symbol calculus for the C*-algebra and a Fredholm criterion for its elements are obtained. For the C*-algebra composed by all functional operators in , an invertibility criterion for its elements is also established. Both the C*-algebras and are investigated by using a generalization of the local-trajectory method for C*-algebras associated with C*-dynamical systems which is based on the notion of spectral measure. Submitted: April 30, 2007. Accepted: November 5, 2007.     

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Commutative C*-Algebras of Toeplitz Operators on the Unit Ball, II. Geometry of the Level Sets of Symbols

In the first part [16] of this work, we described the commutative C*-algebras generated by Toeplitz operators on the unit ball whose symbols are invariant under the action of certain Abelian groups of biholomorphisms of . Now we study the geometric properties of these symbols. This allows us to prove that the behavior observed in the case of the unit disk (see [3]) admits a natural generalization to the unit ball . Furthermore we give a classification result for commutative Toeplitz operator C*-algebras in terms of geometric and “dynamic” properties of the level sets of generating symbols. This work was partially supported by CONACYT Projects 46936 and 44620, México.     

Absolutely convergent Fourier integrals and classical function spaces

We study the continuity and smoothness properties of functions whose Fourier transforms. belong to , and give sufficient conditions in terms of to ensure that f belongs either to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg(α) and zyg(α) for some 0 < α ≤ 2. These sufficient conditions are also necessary under an additional positivity assumption. Our theorems extend known results from periodic to nonperiodic functions. This research was supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

On Toeplitz-type Operators Related to Wavelets

Let G be the “ax + b”-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on . The structure of the space of Calderón (wavelet) transforms inside is described. Using this result some representations, properties and the Wick calculus of the Calderón-Toeplitz operators T _α acting on whose symbols a = a(ζ) depend on for are investigated. This paper was supported by Grant VEGA 2/0097/08.     

Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C _n depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L ^∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.     

Boundedness and Fredholmness of Pseudodifferential Operators in Variable Exponent Spaces

We prove a statement on the boundedness of a certain class of singular type operators in the weighted spaces with variable exponent p(x) and a power type weight w, from which we derive the boundedness of pseudodifferential operators of H?rmander class S ⁰ _1,0 in such spaces. This gives us a possibility to obtain a necessary and sufficient condition for pseudodifferential operators of the class OPS ^m _1,0 with symbols slowly oscillating at infinity, to be Fredholm within the frameworks of weighted Sobolev spaces with constant smoothness s, variable p(·)-exponent, and exponential weights w. Supported by CONACYT Project No.43432 (Mexico), the Project HAOTA of CEMAT at Instituto Superior Técnico, Lisbon (Portugal) and the INTAS Project “Variable Exponent Analysis” Nr.06-1000017-8792.     

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ^*-algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space on the unit disk in . We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. To the memory of G. S. Litvinchuk     

Optimal Decompositions of Translations of L 2-Functions

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space . Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space , or equivalent the spectral theory of a unitary representation U of the rank-n lattice in . Starting with a non-zero vector , we look for relations among the vectors in the cyclic subspace in generated by ψ. Since these vectors involve infinite “linear combinations,” the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L ²-independence. This refers to infinite linear combinations of integral translates of a fixed function with l ²-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals. Work supported in part by the U.S. National Science Foundation.     

Rate of convergence to a singular steady state of a supercritical parabolic equation

We study global positive solutions of a supercritical parabolic equation which converge to a steady state that is singular at x = 0. We determine the rate of convergence to the singular steady state in where B _ν(0) is a ball in with the center at the origin and radius ν.      

Commutative <Emphasis Type="Italic">C</Emphasis>*-Algebras of Toeplitz Operators on the Unit Ball,I. Bargmann-Type Transforms and Spectral Representations of Toeplitz Operators

Extending known results for the unit disk, we prove that for the unit ball there exist n+2 different cases of commutative C*-algebras generated by Toeplitz operators, acting on weighted Bergman spaces. In all cases the bounded measurable symbols of Toeplitz operators are invariant under the action of certain commutative subgroups of biholomorphisms of the unit ball. This work was partially supported by CONACYT Projects 46936 and 44620, México.     

A transference principle for general groups and functional calculus on UMD spaces

Let –iA be the generator of a C ₀-group on a Banach space X and ω > θ(U), the group type of U. We prove a transference principle that allows to estimate in terms of the -Fourier multiplier norm of . If X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev–de Laubenfels. If X is a UMD space, one obtains a bounded -calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded -calculus on sectors.     

$${\mathcal{R}}$$ -boundedness,pseudodifferential operators,and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class () and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with -bounded symbols, yielding by an iteration argument the -boundedness of λ(A−λ)⁻¹ in for some . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with -bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on with operator valued coefficients.     

A spectral characterization of the uniform continuity of strongly continuous groups

Let X be a Banach space and a strongly continuous group of linear operators on X. Set and where is the unit circle and denotes the spectrum of T(t). The main result of this paper is: is uniformly continuous if and only if is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived. Received: 14 June 2006, Revised: 27 September 2007     

Sums of Bisectorial Operators and Applications

We study sums of bisectorial operators on a Banach space X and show that interpolation spaces between X and D(A) (resp. D(B)) are maximal regularity spaces for the problem Ay + By = x in X. This is applied to the study of regularity properties of the evolution equation u′ + Au = f on for or and the evolution equation u′ + Au = f on [0, 2π] with periodic boundary condition u(0) = u(2π) in or