Estimates for the norms of fractional derivatives in terms of integral moduli of continuity and their applications

For functions defined on the entire real axis or a semiaxis, we obtain Kolmogorov-type inequalities that estimate the L _p -norms (1 ≤ p < ∞) of fractional derivatives in terms of the L _p -norms of functions (or the L _p -norms of their truncated derivatives) and their L _p -moduli of continuity and establish their sharpness for p = 1: Applications of the obtained inequalities are given.     

Conjugation in Representations of the Zassenhaus Algebra

Let F be an algebracially closed field of characteristic p > 2, and L be the p ⁿ -dimensional Zassenhaus algebra with the maximal invariant subalgebra L ₀ and the standard filtration {L _i }| ^pn−2 _{i =−1}. Then the number of isomorphism classes of simple L-modules is equal to that of simple L ₀-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p ⁿ . Received May 10, 1999, Accepted December 8, 1999     

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1

Çavu§  A.  Israfilov  D. M. 《分析论及其应用》1995,11(1):105-118

Jackson and Bernstein theorems for the weight exp(−|x|) on ℝ

In 1978, Freud, Giroux and Rahman established a weightedL ₁ Jackson theorem for the weight exp(−|x|) on the real line, using methods that work only inL ₁. This weight is somewhat exceptional, for it sits on the boundary between weights like exp(-|x|^α), α≥1, where weighted polynomials are dense, and the case α<1, where weighted polynomials are not dense. We obtain the firstL _p Jackson theorem for exp(−|x|), valid in allL _p , 0<p≤∞, as well as for higher order moduli of continuity. We also establish a converse Bernstein type theorem, characterizing rates of approximation in terms of smoothness of the approximated function. Research supported by NSF grant DMS 0400446.     

On the Fréchet space L p −

This paper contains a study of the structure of the Fréchet space L _p ⁻, 1< p ≤∞, defined as the intersection of L _q [0,1] for q<p, and endowed with the projective topology. The main topics covered are: normable, Schwartz and nuclear subspaces of L _p ⁻; construction of uncomplemented copies of ?₂ inside L _p ⁻ for p<2; construction of Montel non-Schwartz subspaces; the space L _p ⁻ is primary. Received: 30 October 1996 / Revised version: 1 February 1998     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

Weak amenability components ofL 1(G)-modules, amenable groups, and an ergodic theorem

Weak versions of amenability components of BanachL ₁(G)-modules are considered. Using these versions, a mean ergodic theorem for locally compact groups is formulated. The possibility of using weak amenability components of operatorL ₁(G)-modules to characterize ameable groups is studied. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 879–886, December, 1999.     

Embedding of Hpω in the class eL

We obtain the necessary conditions for the embedding H _p ^ω ⊂ e ^L (1≤p∞) with convex modulus of continuity ω in terms of this modulus. In the case p=1 these conditions are also sufficient.     

On the galois module structure of semisimple holomorphic differentials

LetL/K be a finite Galoisp-extension of algebraic function fields of one variable over an algebraically closed fieldk of characteristicp, with Galois groupG=Gal(L/K). The space Ώ _L ^s (0) of semisimple holomorphic differentials ofL is thek-vector space of holomorphic differentials which are fixed by the Cartier operator. We obtain the isomorphism classes and multiplicities of the summands in a Krull-Schmidt decomposition of thek[G]-module Ώ _L ^s (0) into a direct sum of indecomposablek[G]-modules. Partially supported by CONACyT, project No. 25063-E.     

Estimate of the approximation of periodic functions by negative Cesàro means

The problem of approximation of continuous functions by Cesàro (C,α)-means, −1 < α < 0, in terms of L _p and C-modulus of continuity is studied.     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

Calmness Properties and Contingent Subgradients of Integral Functionals on Lebesgue Spaces L p , 1 ≼ p < ∞

We study the notions of calmness, contingent epiderivatives, Hadamard differentiability, Gateaux differentiability, and contingent subdifferentials of an integral functional defined on a Lebesgue space L _p , with p ≠ ∞.     

On multiplicative perturbations of C 0 -groups and C 0 -cosine operator functions

Certain convolution operators of the form (K f) (t) = A∈t ₀ ^t L(t-s) f(s) ds , where A is the infinitesimal generator of either a C ₀ -group or a C ₀ -cosine family in a Banach space E , are considered. We obtain several lifting results guaranteeing that the continuity of K from L ^p to L ^q implies the continuity of K from L ^p to L ^∈ fty . These results are applied to the study of multiplicative perturbations of C ₀ -groups and C ₀ -cosine families in Banach spaces and to the study of the Maximal Regularly Property (MRP) in L ^p , 1 ≤ p ≤ +∈fty , for second-order Cauchy problem. It is proved that the MRP is equivalent to the boundedness of the infinitesimal generator. April 30, 1999     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (\mathbbZ/l\mathbbZ)^m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.     

On some extensions of <Emphasis Type="Italic">p</Emphasis>-restricted completely splittable GL(<Emphasis Type="Italic">n</Emphasis>)-modules

In this paper, we calculate the space Ext _GL(n ¹ )(L _n (λ), L _n (μ)), where GL(n) is the general linear group of degree n over an algebraically closed field of positive characteristic, L _n (λ) and L _n (μ) are rational irreducible GL(n)-modules with highest weights λ and μ, respectively, the restriction of L _n (λ) to any Levi subgroup of GL(n) is semisimple, λ is a p-restricted weight, and μ does not strictly dominate λ. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 219–226, 2005.     

Isomorphism classes and adjoints of certain iwasawa modules

Letp be an odd prime number andO the integer ring of a finite extension of ℚ _p . We determine isomorphism classes of certainO[[T]]-modules which are isomorphic toO ^⊕3 asO-modules. Moreover we give some examples which are not isomorphic to their adjoints. Partly supported by the Grants-in-Aid for Encouragement of Young Scientists (No. 11740020), The Ministry of Education, Science, Sports and Culture of Japan.     

Tensor products of Drinfeld modules andv-adic representations

We define the tensor product ϕ ⊗ ψ and relatedt-modules Sym²(ϕ), and ∧²(ϕ) for Drinfeld modules ϕ, ψ defined over the rational function fieldK=F _q (T), and describe thev-adic Tate modules of theset-modules by using those of ϕ, ψ.     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Linear-topological classification of separableL p-spaces associated with von Neumann algebras of type I

We classify, up to a linear-topological isomorphism, all separableL _p-spaces, 1≤p<∞, associated with von Neumann algebras of type I. In particular, anyL _p-space associated with an infinite-dimensional atomic von Neumann algebra is isomorphic tol _p, or toC _p, or to . Further, anyL _p-space,p∈[1,∞),p∈2 associated with an infinite-dimensional von Neumann algebraM of type I is isomorphic to one of the following nine Banach spaces: l_p, L_p, S_P, C_p, S_p ⊕ L_p, L_p(S_p), C_p ⊕ L_p, L_p(C_p), C_p ⊕ L_p(S_p). In the casep=1 all the spaces in this list are pairwise non-isomorphic. Research supported by the Australian Research Council.