Displaying similar documents to “Diagonals of separately continuous functions of n variables with values in strongly σ -metrizable spaces”

Baire one functions and their sets of discontinuity

Jonald P. Fenecios, Emmanuel A. Cabral, Abraham P. Racca (2016)

Mathematica Bohemica

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A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function f : is of the first Baire class if and only if for each ϵ > 0 there is a sequence of closed sets { C n } n = 1 such that D f = n = 1 C n and ω f ( C n ) < ϵ for each n where ω f ( C n ) = sup { | f ( x ) - f ( y ) | : x , y C n } and D f denotes the set of points of discontinuity of f . The proof of the main theorem is based on a recent ϵ - δ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications...

Baire classes of complex L 1 -preduals

Pavel Ludvík, Jiří Spurný (2015)

Czechoslovak Mathematical Journal

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Let X be a complex L 1 -predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire- α functions on the set ext B X * of the extreme points of the dual unit ball B X * to the whole unit ball B X * . As a corollary we show that, given α [ 1 , ω 1 ) , the intrinsic α -th Baire class of X can be identified with the space of bounded homogeneous Baire- α functions on the set ext B X * when ext B X * satisfies certain topological assumptions. The paper is intended to be a complex counterpart to...

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart, Yanick Heurteaux (2012)

Annales scientifiques de l'École Normale Supérieure

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A famous theorem of Carleson says that, given any function f L p ( 𝕋 ) , p ( 1 , + ) , its Fourier series ( S n f ( x ) ) converges for almost every x 𝕋 . Beside this property, the series may diverge at some point, without exceeding O ( n 1 / p ) . We define the divergence index at  x as the infimum of the positive real numbers β such that S n f ( x ) = O ( n β ) and we are interested in the size of the exceptional sets E β , namely the sets of  x 𝕋 with divergence index equal to  β . We show that quasi-all functions in  L p ( 𝕋 ) have a multifractal behavior with respect to...

On strong measure zero subsets of κ 2

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

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We study the generalized Cantor space κ 2 and the generalized Baire space κ κ as analogues of the classical Cantor and Baire spaces. We equip κ κ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of κ 2 . We prove for successor κ = κ < κ that the ideal of strong measure zero sets of κ 2 is κ -additive, where κ is the size of the smallest unbounded family in κ κ , and that the generalized Borel...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...