The Spectral Mapping Property of Delay Semigroups

We offer a new way of proving spectral mapping properties of delay semigroups in L ^p -history spaces with finitely many rationally depending delays based on an explicit construction of approximate eigenvectors. This allows us to provide proper generalizations of the existing spectral mapping theorems. Submitted: April 25, 2007. Accepted: November 5, 2007.     

On convex risk measures on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L ^∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on L ^p spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on L ^p we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for L ^p risks.     

Spaces of p-integrable Functions with Respect to a Vector Measure

We study several properties of the Banach lattices L^p (m) and L^p_w (m) of p-integrable scalar functions and weakly p-integrable scalar functions with respect to a countably additive vector measure m. The relation between these two spaces plays a fundamental role in our analysis. This research has been partially supported by La Consejería de Educatión y Ciencia de la Junta de Andalucía.     

Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies

We provide a representation for strong-weak continuous dynamic risk measures from L^p into L^p_t spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.     

Yan Theorem in L∞ with Applications to Asset Pricing

We prove an L~∞ version of the Yan theorem and deduce from it a necessary condition for theabsence of free lunches in a model of financial markets,in which asset prices are a continuous R~d valued processand only simple investment strategies are admissible.Our proof is based on a new separation theorem for convexsets of finitely additive measures.     

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.     

A New Characterization for Carleson Measures and Some Applications

We provide a new characterization for Carleson measures in terms of the L ^p behaviors of certain functions represented as an integration on a non-tangential cone. Applications for characterizing the boundedness and compactness of Volterra type operators from Hardy spaces to some holomorphic spaces are also presented.     

Finite representability ofl p(X) in Orlicz function spaces

We show that ifl _p(X),p ≠ 2, is finitely crudely representable in an Orlicz spaceL _ϕ (which does not containc ₀) then the Banach spaceX is isomorphic to a subspace ofL _p. The same remains true forp = 2 whenL _ϕ is 2-concave or 2-convex, or ifX has local unconditional structure. We extend a theorem of Guerre and Levy to Orlicz function spaces.     

On the Representation of Band Limited Functions Using Finitely Many Bits

In this paper, we consider the question of representing an entire function of finite order and type in terms of finitely many bits, and reconstructing the function from these. Instead of making any further assumptions about the function, we measure the error in reconstruction in a suitably weighted L^p norm. The optimal number of bits in order to obtain a given accuracy is given by the Kolmogorov entropy. We determine this entropy in the case of certain compact subsets of these weighted L^p spaces and obtain constructive algorithms to determine the asymptotically optimal bit representation from finitely many samples of the function. Our theory includes both equidistant and non-uniform sampling. The reconstructions are polynomials, having several other optimality properties.     

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.      

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.     

The Relationship Between Various Filter Notions on aGL-Monoid

The notion of a generalised filter is extended to the setting of aGL-monoid. It is shown that there exists a one-to-one correspondence between the collection of generalised filters on a setXand the collection of strongly stratifiedL-filters onX. Specialising to the case whereLis the closed unit interval [0, c] viewed as a Heyting algebra, we show that any strongly stratified [0, c]-filter onXcan be uniquely identified with a saturated filter onI^Xwith characteristic valuec. In this way, the notion of a generalised filter unifies various filter notions. In particular, necessity measures and finitely additive probability measures are specific examples of generalised filters.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

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.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

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.     

Best Local Approximation in Orlicz Spaces

We get results in Orlicz spaces L ^φ about best local approximation on non-balanced neighborhoods when φ satisfies a certain asymptotic condition. This fact generalizes known previous results in L ^p spaces.     

On the jackson theorem for periodic functions in spaces with integral metric

We consider the approximation of periodic functions by trigonometric polynomials in metric (not normed) spaces that are generalizations of the spaces L _p , 0 < p < 1, and L ₀. In particular, we prove the multidimensional Jackson theorem in L _p (T ^m ), 0 < p < 1.     

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.     

Finitely purely atomic measures: Coincidence and rigidity properties

This paper is a natural continuation of the paper [2] by the same author. We shall prove that several coincidence and rigidity phenomena which usually do not appear are possible only in case the underlying measure space is trivial (i.e. is a finite union of atoms). Examples: coincidence of twoL ^p spaces, reflexivity ofL ¹, Radon—Nikodym property ofL ^∞, coincidence of Dunford, Pettis or Bochner integrability, coincidence of theL ^p space and of the weakL ^p space.