Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

Embedding Theorems for Hölder Classes Defined on p-Adic Linear Spaces

We prove analogues of P. L. Ul’yanov and V. A. Andrienko results concerning embeddings of L^q Hölder spaces into Lebesgue spaces L^r or L^r Hölder spaces in the case 1 ≤ q < r ≤ 2 for functions defined on p-adic linear spaces. The conditions presented in these theorems are sharp. Also we give necessary and sufficient conditions for such embeddings in the case 1 ≤ q < r < ∞ that generalize recent results of S. S. Platonov.

     

A characterization of the interpolation spaces ofH 1 andL ∞ on the line

The Calderón-Mitjagin theorem characterizes all interpolation spaces of the pair of Lebesgue spaces (L ¹,L ) as the rearrangement-invariant spaces. The results of this paper show that the interpolation spaces ofH ¹(R) andL (R) consist of elements whose nontangential maximal functions lie in rearrangement-invariant spaces.Communicated by Jaak Peetre.     

Sur un théorème de J. L. Krivine concernant la caracterisation des classes d’espaces isomorphes a des espaces d’orlicz généralisés et des classes voisines

The classL ^p the classes λ-SL ^p of spaces λ isomorphic to subspaces ofL ^p spaces and λ-QL ^p of subquotients have been characterized in the literature by formulas of certain simple forms. A theorem of Krivine gives a general demonstration of these results in the framework of ψ normed spaces. In particular, characterizations of subspaces and subquotients of certain classes of generalized Orlicz spaces are obtained.      

Natural examples of Valdivia compact spaces

We collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L¹ spaces, Banach lattices and noncommutative L¹ spaces.     

Multilinear Singular and Fractional Integrals

In this paper, we treat a class of non-standard commutators with higher order remainders in the Lipschitz spaces and give （L^v, L^q）, （H^p, L^q） boundedness and the boundedness in the Triebel- Lizorkin spaces. Our results give simplified proofs of the recent works by Chen, and extend his result.     

Adjoining an Order Unit to a Matrix Ordered Space

We prove that an order unit can be adjoined to every L ^∞-matricially Riesz normed space. We introduce a notion of strong subspaces. The matrix order unit space obtained by adjoining an order unit to an L ^∞-matrically Riesz normed space is unique in the sense that the former is a strong L ^∞-matricially Riesz normed ideal of the later with codimension one. As an application of this result we extend Arveson’s extension theorem to L ^∞-matircially Riesz normed spaces. As another application of the above adjoining we generalize Wittstock’s decomposition of completely bounded maps into completely positive maps on C ^*-algebras to L ^∞-matricially Riesz normed spaces. We obtain sharper results in the case of approximate matrix order unit spaces. Mathematics Subject Classification (2000). Primary 46L07     

Equazioni quasi ellittiche e spaziL p, θ (ω, δ) (I)

Summary Regularity theorems inL ^{2, θ} (ω, δ) spaces are proved for weak solutions of quasielliptic differential equations. In particular, regularization results are obtained in the class of holder continuous functions (with respect to a suitable metric related to the operator). As a consequence, we obtain results and estimates in L^p andL ^{p, θ} spaces for the solution of the Dirichlet problem.

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Lavoro eseguito nell’ambito del Gruppo di Ricerca n^o 46 del Comitato per la Matematica del C N.R.     

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L¹‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L¹‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L¹‐spaces. For soft potentials, in L¹‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.     

Error estimates for mixed finite element methods for nonlinear parabolic problems

We consider a class of mixed finite element methods for nonlinear parabolic problems over a plane domain. The finite element spaces taken are Raviart-Thomas spaces of index k, k ? 0. We obtain optimal order L²- and almost optimal order L^∞-error estimates for the finite element solution and order optimal L²-error estimates for its gradient. We also derive the error estimates for the time derivatives of the solution. Our results extend those previously obtained by Johnson and Thomée for the corresponding linear problems with k ? 1.     

Approximate identities in variable L p spaces

We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable L ^p spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable L ^p spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Harmonic analysis on non-semisimple symmetric spaces

It is proven that theL ² spectrum for certain non-semisimple, non-nilpotent symmetric spaces is multiplicity-free. The spectrum and spectral measure are computed precisely for symmetric spaces corresponding to non-compact motion groups. Somewhat less complete results on theL ² spectrum — in both the Mackey Machine and Orbit Method modes — are given for general semidirect product symmetric spaces. The author was supported by the NSF through DMS84-00900-A01 and by a Senior Fulbright Fellowship.     

A damped hyerbolic equation with critical exponent

The aim of this paper is twofold. First, we initiate a detailed study of the so-called X^s _θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and L^p (L^q ) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new L^p → L^q Carleman estimates, derived using the X^s _θ spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that characteristic set satisfies a curvature conditions.     

The Lp - Lq Estimates for the Wave Equation with A nonnegative Potential

In this paper, some Fredholm properties of the Dirichlet problem for the Laplace operator in an exterior domain are given in weighted Hölder spaces. These results extend the corresponding approach in weighted L ^p -Sobolev spaces.     

Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the L ^p -compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L ^p -distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincaré inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.      

Local geometry of L 1 ν L ∞ and L 1+L ∞

Extreme points of the unit sphere S (L ¹+L ^∞ ) of LL ¹+L ^∞ under the classical norm used in the interpolation theory were characterized in [8] and [11], while extreme points of S(L ¹ ∩ L ^∞) under the classical norm were characterized in [7]. In this paper extreme points of the unit sphere of L ¹+L ^∞ and L ¹ ∩ L ^∞ under the “dual” norms are characterized. Moreover, all the extreme points in L ¹ ∩ L ^∞ and L ¹+L ^∞ (under both kinds of norms) are examined if they are exposed, H-points, or strongly exposed. Smooth points in both these spaces for both the norms are also characterized. Finally, it is proved that in general the spaces L ^p +L ^q and L ^p ∩ L ^q are not isometric to Orlicz spaces, provided that 1<p<q<+∞. The paper was written while the first named author was visiting The University of Memphis The third named author is supported by KBN-Grant 2 PO3A 050 09.     

Hardy spaces for α-harmonic functions in regular domains

We investigate Hardy spaces H ^p for singular α-harmonic functions in bounded domains with regular boundaries. We show the correspondence between these spaces and suitable L ^p spaces and measure spaces.     

Properties of representations of operators acting between spaces of vector-valued functions

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L ¹-spaces into L ^∞-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L ^p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.     

Groups of Isometries of Hardy-Type Spaces of Analytic Functionals in the Unit Ball of the Space~L p

The tensor structure of spaces L _p (R ⁿ ) of summable functions of several variables is described. A scale of Hardy-type spaces of analytic functionals defined in the unit ball of the space L _p (R ¹) of summable functions of one variable is introduced. One-parameter groups of isometries of such spaces of analytic functionals are investigated.     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.