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1.

Astashkin, Sergey

et al.

Samara State University.

Maligranda, Lech

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.

Geometry of Cesaro function spaces2011In: Functional analysis and its applications, ISSN 0016-2663, E-ISSN 1573-8485, Vol. 45, no 1, p. 64-68Article in journal (Refereed)

Abstract [en]

Geometric properties of CesA ro function spaces Ces (p) (I), where I = [0,a) or I = [0, 1], are investigated. In both cases, a description of their dual spaces for 1 < p < a is given. We find the type and the cotype of CesA ro spaces and present a complete characterization of the spaces l (q) that have isomorphic copies in Ces (p) [0, 1] (1 a (c) 1/2 p < a).

Over algebraically closed fields of characteristic p > 2, -prolongations of simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. We discover several new simple Lie superalgebras, serial and exceptional, including super versions of Brown and Melikyan algebras, and thus corroborate the super analog of the Kostrikin-Shafarevich conjecture. Simple Lie superalgebras with 2 x 2 Cartan matrices are classified.

We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic.

Let T be a singular integral operator, and let 0 < α < 1. If t > 0 and the functions f and Tf are both integrable, then there exists a function g ε BLipα(ct)) such that ∥f - g∥≤ Cdist L1 (f, BLipα (t)and ∥Tf-Tg ∥ L 1 ≤ c ∥ f-g ∥ L1 +dist L1 (Tf B Lip α(t)). (Here B X (τ) is the ball of radius τ and centered at zero in the space X; the constants C and c do not depend on t and f.) The function g is independent of T and is constructed starting with f by a nearly algorithmic procedure resembling the classical Calderón-Zygmund decomposition.

We prove new point-wise inequalities involving the gradient of a function u is an element of C-1(R-n), the modulus of continuity w of the gradient delu, and a certain maximal function M(lozenge)u and show that these inequalities are sharp. A simple particular case corresponding to n = 1 and w(r) = r is the Landau type inequality [GRAPHIC] where the constant 8/3 is best possible and [GRAPHIC]

7.

Shadrin, Sergei

Stockholm University, Faculty of Science, Department of Mathematics.

Some relations for Hurwitz numbers2005In: Functional analysis and its applications, ISSN 0016-2663, E-ISSN 1573-8485, Vol. 39, no 02, p. 160-162Article in journal (Refereed)

Abstract [en]

We obtain new relations for Hurwitz numbers of functions with one polynomial and one arbitrary critical value. (All other critical values are supposed to be simple.) This is a straightforward generalization of our earlier results on Hurwitz numbers of functions with one polynomial critical value.