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Displaying similar documents to “Generalized α-variation and Lebesgue equivalence to differentiable functions”

Approximate and L p Peano derivatives of nonintegral order

J. Marshall Ash, Hajrudin Fejzić (2005)

Studia Mathematica

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Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. L p , 1 ≤ p ≤ ∞) sense at x m if there are numbers f α ( x ) , |α| ≤ n, such that f ( x + h ) - | α | n f α ( x ) h α / α ! is O ( h u ) in the approximate (resp. L p ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L p sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...

Fréchet differentiability via partial Fréchet differentiability

Luděk Zajíček (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let X 1 , , X n be Banach spaces and f a real function on X = X 1 × × X n . Let A f be the set of all points x X at which f is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if X 1 , , X n - 1 are Asplund spaces and f is continuous (respectively Lipschitz) on X , then A f is a first category set (respectively a σ -upper porous set). We also prove that if X , Y are separable Banach spaces and f : X Y is a Lipschitz mapping, then there exists a σ -upper porous set A X such that f is Fréchet differentiable...

On compactness and connectedness of the paratingent

Wojciech Zygmunt (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this note we shall prove that for a continuous function ϕ : Δ n , where Δ ,  the paratingent of ϕ at a Δ is a non-empty and compact set in n if and only if ϕ satisfies Lipschitz condition in a neighbourhood of a . Moreover, in this case the paratingent is a connected set.

Multiple conjugate functions and multiplicative Lipschitz classes

Ferenc Móricz (2009)

Colloquium Mathematicae

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We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class L i p ( α , . . . , α N ) for some 0 < α , . . . , α N < 1 and its marginal functions satisfy f ( · , x , . . . , x N ) L i p β , . . . , f ( x , . . . , x N - 1 , · ) L i p β N for some 0 < β , . . . , β N < 1 uniformly in the indicated variables x l , 1 ≤ l ≤ N, then f ̃ ( η , . . . , η N ) L i p ( α , . . . , α N ) for each choice of ( η , . . . , η N ) with η l = 0 or 1 for 1 ≤ l ≤ N.

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 .

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that real functions on which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower C 1 -functions, or of two strongly paraconvex functions) coincide with semismooth functions on (i.e. those locally Lipschitz functions on for which f + ' ( x ) = lim t x + f + ' ( t ) and f - ' ( x ) = lim t x - f - ' ( t ) for each x ). Further, for each modulus ω , we characterize the class D S C ω of functions on which can be written as f = g - h , where g and h are semiconvex with modulus C ω (for some C > 0 ) using a new...

Density of smooth maps for fractional Sobolev spaces W s , p into simply connected manifolds when s 1

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2013)

Confluentes Mathematici

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Given a compact manifold N n ν and real numbers s 1 and 1 p &lt; , we prove that the class C ( Q ¯ m ; N n ) of smooth maps on the cube with values into N n is strongly dense in the fractional Sobolev space W s , p ( Q m ; N n ) when N n is s p simply connected. For s p integer, we prove weak sequential density of C ( Q ¯ m ; N n ) when N n is s p - 1 simply connected. The proofs are based on the existence of a retraction of ν onto N n except for a small subset of N n and on a pointwise estimate of fractional derivatives of composition of maps in W s , p W 1 , s p .

L p - L q boundedness of analytic families of fractional integrals

Valentina Casarino, Silvia Secco (2008)

Studia Mathematica

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We consider a double analytic family of fractional integrals S z γ , α along the curve t | t | α , introduced for α = 2 by L. Grafakos in 1993 and defined by ( S z γ , α f ) ( x , x ) : = 1 / Γ ( z + 1 / 2 ) | u - 1 | z ψ ( u - 1 ) f ( x - t , x - u | t | α ) d u | t | γ d t / t , where ψ is a bump function on ℝ supported near the origin, f c ( ² ) , z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that S z γ , α maps L p ( ² ) to L q ( ² ) boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel K - 1 + i θ i ϱ , α is a product kernel on ℝ², adapted to the curve t | t | α ; as a consequence, we show...

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...

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.

Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen (2004)

Studia Mathematica

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For 0 < γ ≤ 1, let Λ γ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let J γ ( E , Q ) be the closed ideal in Λ γ consisting of those functions f Λ γ for which (i) f = 0 on E, (ii) | f ( z ) - f ( w ) | = o ( | z - w | γ ) as d(z,E),d(w,E) → 0, (iii) f / Q Λ γ . Also, for a closed ideal I in Λ γ , let E I = z ∈ : f(z) = 0 for every f ∈ I and let Q I be the greatest common divisor of the inner parts of non-zero functions...

Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

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Let E be a compact set in the complex plane, g E be the Green function of the unbounded component of E with pole at infinity and M ( E ) = s u p ( | | P ' | | E ) / ( | | P | | E ) where the supremum is taken over all polynomials P | E 0 of degree at most n, and | | f | | E = s u p | f ( z ) | : z E . The paper deals with recent results concerning a connection between the smoothness of g E (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence M ( E ) n = 1 , 2 , . . . . Some additional conditions are given for special classes of sets.

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space ( X , d , μ ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B ( x , r ) converge to f ( x ) when r converges to 0 . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

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 .

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

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We consider the equation - y ' ( x ) + q ( x ) y ( x - ϕ ( x ) ) = f ( x ) , x , where ϕ and q ( q 1 ) are positive continuous functions for all x and f C ( ) . By a solution of the equation we mean any function y , continuously differentiable everywhere in , which satisfies the equation for all x . We show that under certain additional conditions on the functions ϕ and q , the above equation has a unique solution y , satisfying the inequality y ' C ( ) + q y C ( ) c f C ( ) , where the constant c ( 0 , ) does not depend on the choice of f .

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

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We consider the integral functional J ( u ) = Ω [ f ( | D u | ) u ] d x , u W 0 1 , 1 ( Ω ) , where Ω n , n 2 , is a nonempty bounded connected open subset of n with smooth boundary, and s f ( | s | ) is a convex, differentiable function. We prove that if J admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball.

Hardness of embedding simplicial complexes in d

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let 𝙴𝙼𝙱𝙴𝙳 k d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k , does there exist a (piecewise linear) embedding of K into d ? Known results easily imply polynomiality of 𝙴𝙼𝙱𝙴𝙳 k 2 ( k = 1 , 2 ; the case k = 1 , d = 2 is graph planarity) and of 𝙴𝙼𝙱𝙴𝙳 k 2 k for all k 3 . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that 𝙴𝙼𝙱𝙴𝙳 d d and 𝙴𝙼𝙱𝙴𝙳 ( d - 1 ) d are undecidable for each d 5 . Our main result is NP-hardness of 𝙴𝙼𝙱𝙴𝙳 2 4 and, more generally, of 𝙴𝙼𝙱𝙴𝙳 k d for all...

A Note on VLO Functions

Francesca Angrisani, Giacomo Ascione (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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Inspired by a result from Leibov, we find that the supremum defining the B L O norm in [ 0 , 1 ] is actually attained by a specific sub-interval of [ 0 , 1 ] for f V L O ( [ 0 , 1 ] )