Displaying similar documents to “Absolute continuity with respect to a subset of an interval”

Selectors of discrete coarse spaces

Igor Protasov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Given a coarse space ( X , ) with the bornology of bounded subsets, we extend the coarse structure from X × X to the natural coarse structure on ( { } ) × ( { } ) and say that a macro-uniform mapping f : ( { } ) X (or f : [ X ] 2 X ) is a selector (or 2-selector) of ( X , ) if f ( A ) A for each A { } ( A [ X ] 2 , respectively). We prove that a discrete coarse space ( X , ) admits a selector if and only if ( X , ) admits a 2-selector if and only if there exists a linear order “ " on X such that the family of intervals { [ a , b ] : a , b X , a b } is a base for the bornology .

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

On almost everywhere differentiability of the metric projection on closed sets in l p ( n ) , 2 < p <

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

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Let F be a closed subset of n and let P ( x ) denote the metric projection (closest point mapping) of x n onto F in l p -norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in n in the Euclidean case p = 2 . We consider the case 2 < p < and prove that the i th component P i ( x ) of P ( x ) is differentiable a.e. if P i ( x ) x i and satisfies Hölder condition of order 1 / ( p - 1 ) if P i ( x ) = x i .

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...

The basic construction from the conditional expectation on the quantum double of a finite group

Qiaoling Xin, Lining Jiang, Zhenhua Ma (2015)

Czechoslovak Mathematical Journal

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Let G be a finite group and H a subgroup. Denote by D ( G ; H ) (or D ( G ) ) the crossed product of C ( G ) and H (or G ) with respect to the adjoint action of the latter on the former. Consider the algebra D ( G ) , e generated by D ( G ) and e , where we regard E as an idempotent operator e on D ( G ) for a certain conditional expectation E of D ( G ) onto D ( G ; H ) . Let us call D ( G ) , e the basic construction from the conditional expectation E : D ( G ) D ( G ; H ) . The paper constructs a crossed product algebra C ( G / H × G ) G , and proves that there is an algebra isomorphism between...

Complex series and connected sets

B. Jasek

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CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO Σ 1 , Σ 2 , Σ 3 , Σ 4 INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

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...

Norm continuity of pointwise quasi-continuous mappings

Alireza Kamel Mirmostafaee (2018)

Mathematica Bohemica

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Let X be a Baire space, Y be a compact Hausdorff space and ϕ : X C p ( Y ) be a quasi-continuous mapping. For a proximal subset H of Y × Y we will use topological games 𝒢 1 ( H ) and 𝒢 2 ( H ) on Y × Y between two players to prove that if the first player has a winning strategy in these games, then ϕ is norm continuous on a dense G δ subset of X . It follows that if Y is Valdivia compact, each quasi-continuous mapping from a Baire space X to C p ( Y ) is norm continuous on a dense G δ subset of X .

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C ( X ) ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., C ( Y ) ), and ϕ a linear isometry from M into C(Y) (resp., C ( Y ) ). We show, under the assumption that Π N Π T , where Π N is...