Fractional Differences and Lizorkin--Triebel Spaces

In this paper, we obtain inequalities for maximal functions of fractional differences and fractional derivatives and use them to prove that in the Lizorkin--Triebel spaces F^s _pq=F^s _pq(ⁿ), 0 < p < , 0 q , given sufficient smoothness s > 0, we can introduce equivalent quasinorms expressible in terms of fractional differences of order > s.     

Multipliers and Cyclic Vectors in Bloch Type Spaces

In this paper we study the coefficient multipliers, pointwise multipliers and cyclic vectors in the Bloch type spaces B ^α and little Bloch type spaces for 0 < α < ∞. We give several full characterizations of the coefficient multipliers (B ^α ,B ^β ) and for 0 < α, β < ∞ and pointwise multipliers M(B ^α ,B ^β ) and for 1 ≠ α, β ∈ (0,∞). We also obtain some properties of cyclic vectors for Bloch type spaces. Dedicated to Professor Yu Zan HE on the occasion of his 65^th birthday     

Function spaces of varying smoothness I

This paper deals with function spaces of varying smoothness. It is a modified version of corresponding parts of [8]. Corresponding spaces of positive smoothness s (x) will be considered in part II. We define the spaces B_p (?ⁿ ), where the function ??: x ? s (x) is negative and determines the smoothness pointwise. First we prove basic properties and then we use different wavelet decompositions to get information about the local smoothness behavior. The main results are characterizations of the spaces B_p (?ⁿ ) by weighted sequence space norms of the wavelet coefficients. These assertions are used to prove an interesting connection to the so‐called two‐microlocal spaces C^{s,s ′} (x⁰). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multipliers on the Set of Rademacher Series in Symmetric Spaces

Let E be a symmetric space on [0,1]. Let (,E) be the space of measurable functions f such that fg E for every almost everywhere convergent series g=b _n r _n E, where (r _n) are the Rademacher functions. It was shown that, for a broad class of spaces E, the space (,E) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on E for (,E) to be order isomorphic to L . This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which (,E) is order isomorphic to a symmetric space that differs from L . The answer is positive for the Orlicz spaces E=L _q with _q(t)=exp|t|^q-1 and 0相似文献   

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces

It is well known, that for the sums of i.i.d. random variables we have S _n/n 0 a.s. iff _n=1 1/n P(|S _n| > n) < holds for all > 0 (Spitzer's SLLN). The result is also known in separable Banach spaces. It will be shown, that this also holds in nonseparable (= not necessarily separable) Banach spaces without any measurability assumption. In the theory of empirical processes this gives a characterization of Glivenko-Cantelli classes.     

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

Atomic Decomposition of Hardy Spaces Associated with Certain Laguerre Expansions

Let L _n ^a (x), n=0,1,…, be the Laguerre polynomials of order a>−1. Denote ℓ _n ^a (x)=(n!/Γ(n+a+1))^1/2 L _n ^a (x)e ^−x/2. Let

Some Groups Having Only Elementary Actions on Metric Spaces with Hyperbolic Boundaries

We study isometric actions of certain groups on metric spaces with hyperbolic-type bordifications. The class of groups considered includes SL _n (), Artin braid groups and mapping class groups of surfaces (except the lower rank ones). We prove that in various ways such actions must be elementary. Most of our results hold for non-locally compact spaces and extend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also show that SL _n () for n 3 cannot act on a visibility space X without fixing a point in . Corollaries concern Floyd's group completion, linear actions on strictly convex cones, and metrics on the moduli spaces of compact Riemann surfaces. Some remarks on bounded generation are also included.     

Piecewise Linear Bases and Besov Spaces on Fractal Sets

For a class of closed sets F R ⁿ admitting a regular sequence of triangulations or generalized triangulations, the analogues on F of the Faber—Schauder and Franklin bases are discussed. The characterizations of the Besov spaces on F in the terms of coefficients of functions with respect to these bases are proved. As a consequence, analogous characterizations of the Besov spaces on some fractal domains (including the Sierpinski gasket and the von Koch curve) by coefficients of functions with respect to the wavelet bases constructed in [26] are obtained.     

Barrelledness of Generalized Sums of Normed Spaces

Let (E _i ) _iI be a family of normed spaces and a space of scalar generalized sequences. The -sum of the family (E _i ) _iI of spaces is

Beyond Besov Spaces Part 1: Distributions of Wavelet Coefficients

We determine which information can be extracted from the distributions of the wavelet coefficients of a function f at each scale, but does not depend on the particular wavelet basis which is chosen. This information can be naturally expressed in terms of one increasing function _f (), and the knowledge of this function yields strictly more information than the knowledge of the Besov spaces that contain f . Examples of use of this additional information will be taken from image processing and multifractal analysis.     

Wavelets and Geometric Structure for Function Spaces

Abstract With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel–Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L ¹ are studied. Supported by NNSF of China (Grant No. 10001027)     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

Sobolev Type Spaces on the Dual of the Laguerre Hypergroup

Sobolev type spaces E ^s,p (0, sR, p[1,+]) are defined on R×N by using the Fourier transform and its inverse on the Laguerre hypergroup. An analogue of H ^s (R ⁿ ), denoted by H ^s is investigated in this paper. Some properties including completeness and imbedding results for these spaces are given, Reillich-type theorem and Poincaré's inequality are proved. Also, global regularity results for certain differential operators are obtained.     

Weighted Bergman Spaces and Carleson Measures on the Unit Ball of Cn

Let be a normal function on [0, 1), B _n the unit ball of C ⁿ , and A ^p (B _n ) the weighted Bergman spaces on B _n with weight . The purpose of this paper is to discuss some relations among A ^p (B _n ), weighted Bergman kernels, and Carleson measures on B _n .     

Function Spaces with Exponential Weights II

This paper is a continuation of [8]. We study weighted function spaces of type B and F on the Euclidean space Rⁿ, where u is a weight function of at most exponential growth. In particular, u(χ (±|χ|) is an admissible weight. We deal with atomic decompositions of these spaces. Furthermore, we prove that the spaces B and F are isomorphic to the corresponding unweighted spaces B and F.     

Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

We review the proposal of a constructive axiomatic approach to the determination of the orbit spaces of all the real compact linear groups, obtained through the computation of a metric matrix , which is defined only in terms of the scalar products between the gradients p₁(x),...,p_q(x) of the elements of a minimal integrity basis (MIB) for the ring [ⁿ]^G of G-invariant polynomials. The domain of semi-positivity of is known to realize the orbit space ⁿ/G of G as a semi-algebraic variety in the space ^q spanned by the variables p₁,...,p_q. The matrices can be obtained from the solutions of a universal differential equation (master equation), which satisfy convenient initial conditions. The master equation and the initial conditions involve as free parameters only the degrees d_a of the p_a(x)s. This approach tries to bypass the actual impossibility of explicitly determining a set of basic polynomial invariants for each group. Our results may be relevant in physical contexts where the study of covariant or invariant functions is important, like in the determination of patterns of spontaneous symmetry breaking in quantum field theory, in the analysis of phase spaces and structural phase transitions (Landaus theory), in covariant bifurcation theory, in crystal field theory and so on. Mathematics Subject Classifications (2000) 14L24, 13A50, 14L30.This paper is partially supported by INFN and MURST 40% and 60%.