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Displaying similar documents to “Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist”

The topology of the space of ℋ𝒦 integrable functions in n

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

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It is known that there is no natural Banach norm on the space ℋ𝒦 of n -dimensional Henstock-Kurzweil integrable functions on [ a , b ] . We show that the ℋ𝒦 space is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . On each ℋ𝒦 ( X ) space, an F -norm · X is defined. A · X -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F -norm is also defined for a · X -continuous linear operator. Hence, many important results in functional analysis hold for the ℋ𝒦 ( X ) space. It is well-known that every...

Compact operators and integral equations in the ℋ𝒦 space

Varayu Boonpogkrong (2022)

Czechoslovak Mathematical Journal

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The space ℋ𝒦 of Henstock-Kurzweil integrable functions on [ a , b ] is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . In this paper, on each Fréchet space ℋ𝒦 ( X ) , an F -norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the ℋ𝒦 ( X ) space. It is known that every control-convergent sequence in the ℋ𝒦 space always belongs to a ℋ𝒦 ( X ) space for some X . We illustrate how...

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.

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...

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n &gt; 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

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The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then a sequence x A , where A is a finite alphabet, is both α -substitutive and β -substitutive if and only if x is ultimately...

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 .

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...

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...