Smooth points in some spaces of bounded operators

We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel ^p -orc ₀-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y).     

Integral representations of functions and embeddings of sobolev spaces on cuspidal domains

Some classes of cuspidal domainsG ⊂ ℝ ⁿ are introduced, and embeddings of the formW _p ^(l) (G)↪L_q(G),l ∈ ℕ, for sobolev spaces are established. To this end, estimates of some integral operators are needed. These operators cannot be estimated via Riesz potentials or their anisotropic analogs. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 201–219, February, 1997. Translated by V. E. Nazaikinskii     

Bernstein widths of embeddings of Lebesgue spaces

Let (K, μ) be a measurable space with μ(K)=1. Let Ip,q: L^p (K, μ)→L^q (K, μ) be the embedding operator. The Bernstein widths of I_{p, q} are considered. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 166–169. Translated by S. V. Kislyakov.     

Extremum problems in minimax signal detection forl q -ellipsoids with anl p -ball removed

An asymptotic minimax problem of signal detection for signals froml _q -ellipsoids with an l_p-ball removed (p>2 or q<p<2) in a Gaussian white noise is considered. Asymptotically sharp distinguishability conditions and some estimates of the minimax probability of errors of signal detection for ellipsoids with axes a_n≈n^−λ λ>0, are obtained. The proofs use results and techniques developed by Yu. I. Ingster. Bibliography:6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 312–332     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Analogs of the jackson-nikol'skii inequalities for trigonometric polynomials in spaces with nonsymmetric norm

The paper is concerned with the evaluation of where ▮·▮p ₁, p₂ is a nonsymmetric norm. The order of this number is obtained. Lower bounds involve new polynomials whose properties are studied in detail. In the casep ₁=p ₂,q ₁=q ₂, the estimate obtained is reduced to the well-known Jackson-Nikol'skii inequality. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 687–699, May, 1997. Translated by N. K. Kulman     

On an inclusion between operator ideals

Let 1 ⩽ q < p < ∞ and 1/r:= 1/p max(q/2, 1). We prove that L _r,p ^(c), the ideal of operators of Gel’fand type l _r,p , is contained in the ideal Π _p,q of (p, q)-absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space.     

Inequalities involving upper bounds for certain matrix operators

In this paper, we considered the problem of finding the upper bound Hausdorff matrix operator from sequence spaces l_p(v) (ord(v, p)) intol _p (w) (ord(w, p)). Also we considered the upper bound problem for matrix operators fromd(v, 1) intod(w, 1), and matrix operators frome(w, ∞) intoe(v, ∞), and deduce upper bound for Cesaro, Copson and Hilbert matrix operators, which are recently considered in [5] and [6] and similar to that in [10].     

Dirac operators on the quantum group SU(2) and the quantum sphere

The problem of constructing the Dirac operators on the quantum groupSU(2) and the quantum sphere S_qμ ² are discussed. In both cases, the constructions presented have the sameSU _q(2)-invariant form and are directly connected with the corresponding Laplace operators. Bibliography:16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 49–65. Translated by P. N. Bibikov.     

B-convexity of the space of 2-summing operators

If π₂ (E, F), the class of 2-summing operators fromE intoF, isB-convex, then bothE andF have cotypeq for allq>2. On the other hand, ifE is super-reflexive and has cotype 2, then π₂(l ₂,E) is super-reflexive.     

Contact Elements on Fibered Manifolds

For every product preserving bundle functor T ^μ on fibered manifolds, we describe the underlying functor of any order (r, s, q), s ≥ r ≤ q. We define the bundle K_k,l^r,s,q YK_{k,l}^{r,s,q} Y of (k, l)-dimensional contact elements of the order (r, s, q) on a fibered manifold Y and we characterize its elements geometrically. Then we study the bundle of general contact elements of type μ. We also determine all natural transformations of K_k,l^r,s,q YK_{k,l}^{r,s,q} Y into itself and of T( K_k,l^r,s,q Y )T\left( {K_{k,l}^{r,s,q} Y} \right) into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms from Y to K_k,l^r,s,q YK_{k,l}^{r,s,q} Y .     

Vector-valued Fourier multipliers on symmetric spaces of the noncompact type

LetX be a Riemannian symmetric space of the noncompact type. We prove the multiplier theorem for the Helgason-Fourier transform and the vector valued function spacesL _p (X, l _q ). As a consequence we get the inequalities of the Littlewood-Paley type forL _p (X) spaces.Research supported by K.B.N. Grant 210519101 (Poland).     

A New Class of Perfect Fréchet Spaces

This paper deals with a new class of perfect Frèchet spaces which can be obtained by interpolation of echelon spaces: l_p,q[a_m,n]. We determine the reflexive, Montel, Schwartz, totally reflexive, totally Montel and nuclear spaces l_p,q[a_m,n]. We also derive results on closed subspaces on the spaces (l_p,q)(N).     

Necessary and sufficient conditions for the lipschitzian invertibility of nonlinear difference operators in the spacesl p (ℤ, ℝ) with 1≤p≤∞with 1≤p≤∞

Necessary and sufficient conditions for the Lipschitzian invertibility of the difference map

On properties of the generalized elliptic pseudo-differential operators on closed manifolds

It is shown that two classes of generalized elliptic pseudo-differential operators, GEL(X) and REL(X), selected by the author from the class of classical linear pseudo-differential operators coincide. It is also shown that for any operators A, B ∈GEL(X) their composition AB and their global parametrices P_A, P_B belong to GEL(X). An operator A belongs to GEL(X) independently of the choice of a basis in E and of the weighted order of A. Some properties of the classes EFL(U) and REL(U) arising in microalocal analysis of generalized elliptic operators are studied. Bibliography: 12 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 215–269. Translated by N. A. Karazeeva.     

Quotient normed cones

Given a normed cone (X, p) and a subconeY, we construct and study the quotient normed cone (X/Y,p) generated byY. In particular we characterize the bicompleteness of (X/Y, ‖·‖ ^p ,p) in terms of the bicompleteness of (X, p), and prove that the dual quotient cone ((X/Y)^*, || · ‖·‖^p,p) can be identified as a distinguished subcone of the dual cone (X ^*, || · ||_{p, u}). Furthermore, some parts of the theory are presented in the general setting of the spaceCL(X, Y) of all continuous linear mappings from a normed cone (X, p) to a normed cone (Y, q), extending several well-known results related to open continuous linear mappings between normed linear spaces.     

Quantitative Whitney-type theorems

The crucial point in the construction of an almost optimal decomposition for the couple (L_q, W _p ^k ) was the covering theorem in which the parameter α (α-capacity) was controlled. Here we give a detailed proof of this theorem for negative α. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 96–113. Translated by S. V. Kislyakov.     

One-dimensional Tilings Using Tiles with Two Gap Lengths

Let n,p,k,q,l be positive integers with n=k+l+1. Let x₁,x₂, . . . ,x_n be a sequence of positive integers with x₁<x₂<···<x_n. A set {x₁,x₂, . . . ,x_n} is called a set of type (p,k;q,l) if the set of differences {x₂–x₁,x₃–x₂, . . . ,x_n–x_n–1} equals {p, . . . ,p,q, . . . ,q} as a multiset, where p and q appear k and l times, respectively. Among other results, it is shown that for any p,k,q, there exists a finite interval I in the set of integers such that I is partitioned into sets of type (p,k;q,1).     

Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

Isomorphism criterion for monomial graphs

Let q be a prime power, ??_q be the field of q elements, and k, m be positive integers. A bipartite graph G = G_q(k, m) is defined as follows. The vertex set of G is a union of two copies P and L of two‐dimensional vector spaces over ??_q, with two vertices (p₁, p₂) ∈ P and [ l₁, l₂] ∈ L being adjacent if and only if p₂ + l₂ = pl. We prove that graphs G_q(k, m) and G_q′(k′, m′) are isomorphic if and only if q = q′ and {gcd (k, q ? 1), gcd (m, q ? 1)} = {gcd (k′, q ? 1),gcd (m′, q ? 1)} as multisets. The proof is based on counting the number of complete bipartite INFgraphs in the graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 322–328, 2005