Boundedness of generalized fractional integral operators

The authors introduce a new kind of fractional integral operators, namely, the so called (θ,N)-type fractional integral operators, and discuss their boundedness on the Hardy spaces, the weak Hardy spaces and the Herz-type Hardy spaces.     

Centralisers of spaces of symmetric tensor products and applications

We show that the centraliser of the space of n-fold symmetric injective tensors, n≥2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials. This project was supported in part by Enterprise Ireland, International Collaboration Grant – 2004 (IC/2004/009). The second author was also partially supported by PIP 5272,UBACYTX108 and PICT 03-15033     

New Hardy Spaces Associated with Herz Spaces and Beurling Algebras on Homogeneous Groups

The author introduces the Hardy spaces associated with the Herz spaces and the Beurling algebras on homogeneous groups and establishes their atomic decomposition characterizations. As the applications of this decomposition, the duals of these Hardy spaces and the boundedness of the central δ-Calderón-Zygmund operators on these Hardy spaces are studied. Received January 13, 2000, Accepted July 26, 2000     

Lattice structures and spreading models

We consider problems concerning the partial order structure of the set of spreading models of Banach spaces. We construct examples of spaces showing that the possible structure of these sets include certain classes of finite semi-lattices and countable lattices and all finite lattices. Research of the second named author was partially supported by the National Science Foundation. The third named author had a visiting appointment at the University of South Carolina for the 2004–05 academic year during part of his research.     

Weighted norm inequalities,off-diagonal estimates and elliptic operators Part II: Off-diagonal estimates on spaces of homogeneous type

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant L^p − L^q estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators. This work was partially supported by the European Union (IHP Network “Harmonic Analysis and Related Problems” 2002-2006, Contract HPRN-CT-2001-00273-HARP). The second author was also supported by MEC “Programa Ramón y Cajal, 2005” and by MEC Grant MTM2004-00678.     

New hardy spaces associated with some anisotropic Herz spaces and their applications

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ ⁿ are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces. The research is supported by NSF of China (Grant Nos. 10571014 and 10571015) and SRFDP of China (Grant No. 20050027025)     

Boundedness of Some Maximal Commutators in Hardy-type Spaces with Non-doubling Measures

Let μ be a non-negative Radon measure on R^d which satisfies only some growth conditions. Under this assumption, the boundedness in some Hardy-type spaces is established for a class of maximal Calderón-Zygmund operators and maximal commutators which are variants of the usual maximal commutators generated by Calder6ón- Zygmund operators and RBMO（μ） functions, where the Hardytype spaces are some appropriate subspaces, associated with the considered RBMO（μ） functions, of the Hardv soace H^I（μ） of Tolsa.     

Interpolation for weak Orlicz spaces with M Δ condition

An interpolation theorem for weak Orlicz spaces generalized by N-functions satisfying M _Δ condition is given. It is proved to be true for weak Orlicz martingale spaces by weak atomic decomposition of weak Hardy martingale spaces. And applying the interpolation theorem, we obtain some embedding relationships among weak Orlicz martingale spaces. This work was supported by the National Natural Science Foundation of China (Grant No. 10671147)     

Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and f ∈ L _p(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No. M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No. T043769, T047128, T047132.     

Multilevel decompositions of functional spaces

A unified abstract framework for the multilevel decomposition of both Banach and quasi-Banach spaces is presented. The characterization of intermediate spaces and their duals is derived from general Bernstein and Jackson inequalities. Applications to compactly supported biorthogonal wavelet decompositions of families of Besov spaces are also given. The first author was partially supported by grants from MURST (40% Analisi Numerica) and ASI (Contract ASI-92-RS-89), whereas the second author was partially supported by grants from MURST (40% Analisi Funzionale) and CNR (Progetto Strategico “Applicazioni della Matematica per la Tecnologia e la Società”).     

Hardy Spaces on ? n with Pointwise Variable Anisotropy

In this work we develop highly geometric Hardy spaces, for the full range 0<p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝ ⁿ that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was ‘fixed’ over ℝ ⁿ and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are ‘close’ to 1.     

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant Members of VBAC (Vector Bundles on Algebraic Curves). Second and Third authors partially supported by Ministerio de Educación y Ciencia and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP2002-0017 (Spain)/E–30/03 (Portugal). First and Second authors partially supported by Ministerio de Educación y Ciencia (Spain) through Project MTM2004-07090-C03-01. Third author partially supported by the Centro de Matemática da Universidade do Porto and the project POCTI/MAT/58549/2004, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds. The second author visited the IHES with the partial support of the European Commission through its 6th Framework Programme “Structuring the European Research Area” and the Contract No. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action     

Herz-type Triebel-Lizorkin Spaces, Ⅰ

Let s ∈R,0〈β≤∞, 0〈 q, p〈 ∞ and-n/q〈α. In this paper the authors introduce the Herz-type Triebel-Lizorkin spaces,Kq^α,pFβ^s（R^n）andKq^α,pFβ^s（R^n）which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel-Lizorkin spaces, Some properties on these Herz-type Triebel Lizorkin spaces are also given.     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

Cartan decomposition of the moment map

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.First author partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft and the DFG Schwerpunk program Globale Methoden in der komplexen Geometrie.Second author partially supported by NSA grant H98230–04–01–0070.     

Almost isometric embedding between metric spaces

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU. We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding. The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ. Research of the first author was supported by an Israeli Science foundation grant no. 177/01. Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827.     

Hardy and BMO spaces associated to divergence form elliptic operators

Consider a second order divergence form elliptic operator L with complex bounded measurable coefficients. In general, operators based on L, such as the Riesz transform or square function, may lie beyond the scope of the Calderón–Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some L ^p spaces. In this work we develop a theory of Hardy and BMO spaces associated to L, which includes, in particular, a molecular decomposition, maximal and square function characterizations, duality of Hardy and BMO spaces, and a John–Nirenberg inequality. S. Hofmann was supported by the National Science Foundation.     

Boundedness of commutators on Hardy type spaces

Let [b, T] be the commutator of the function b ∈ Lipβ(Rn) (0 ＜β≤ 1) and the CalderónZygmund singular integral operator T. The authors study the boundedness properties of [b, T] on the classical Hardy spaces and the Herz-type Hardy spaces in non-extreme cases. For the boundedness of these commutators in extreme cases, some characterizations are also given. Moreover, the authors prove that these commutators are bounded from Hardy type spaces to the weak Lebesgue or Herz spaces in extreme cases.     

Linear liftings for non-complete probability spaces

We show that it is consistent with ZFC thatL ^∞(Y,B,ν) has no linear lifting for many non-complete probability spaces (Y,B,ν), in particular forY=[0,1]^A,B=Borel subsets ofY, ν=usual Radon measure onB. Research supported by UPEI Senate Grant no 602101, by the Research Institute for Mathematical Sciences at Bar-Ilan University, and by the Landau Center for Mathematical Research in Analysis (supported by the Minerva Foundation). The author would like to thank the organizers of the Winter Institute on the Set Theory of the Reals for their hospitality while part of this research was being carried out. Partially supported by the Foundation for Basic Research of the Israel Academy of Science. Publication number 437.     

EXTENSIONS OF HILBERT MODULES AND HANKEL OPERATORS

1.IntroductionInthepresentpaper,wewillcontinuethestudyofHilbertmodules[1-5].LetHbeaHilbertspaceandAafunctionalgebra.WesaythatHisaHilbertmoduleoverAifthereisamultiplication(a,f)-affromAxHtoH,makingHintoanXmoduleandif,inaddition,theactionisjointlycontinuousinthesup-normonAandtheHilbertspacenormonH.ThecategoryHofallHilbertNmodulesisanaturalsettingfornumerousquestionsinoperatortheory.In[9],DouglasandPaulseninitiatedasystematicstudyofHilbertmodules'theywereabletotranslatemanyconceptsandpro…