-spaces and -spaces
Nguyen To Nhu (1979)
Colloquium Mathematicae
Similarity:
Nguyen To Nhu (1979)
Colloquium Mathematicae
Similarity:
Ferenc Móricz (2009)
Colloquium Mathematicae
Similarity:
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 for some and its marginal functions satisfy for some uniformly in the indicated variables , 1 ≤ l ≤ N, then for each choice of with or 1 for 1 ≤ l ≤ N.
Alberto Bressan, Agostino Cortesi (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
Similarity:
Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz , definita su un sottoinsieme di uno spazio di Hilbert a valori compatti e convessi in , può essere estesa su tutto ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz .
Ying Xiong, Lifeng Xi (2009)
Studia Mathematica
Similarity:
This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if and are dust-like graph-directed sets satisfying the transitivity condition, then and are Lipschitz equivalent, and and are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.
Witold Marciszewski (2003)
Studia Mathematica
Similarity:
We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight and with the third derived set empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.
Piotr Niemiec (2009)
Studia Mathematica
Similarity:
It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space of diameter r, is (isometrically if r = +∞) isomorphic to the space of equivalence classes of all real-valued Lipschitz maps on . The space of all signed (real-valued) Borel measures on is isometrically embedded in the dual space of and it is shown that the image...
Maria Elena Becker (2005)
Studia Mathematica
Similarity:
Let T be a linear operator on a Banach space X with for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) converges uniformly; (ii) .
Silvano Delladio (2008)
Bollettino dell'Unione Matematica Italiana
Similarity:
We prove a result about the rectifiability of class of the set of regular values (in the sense of Clarke) of a Lipschitz map with
Alberto Bressan, Agostino Cortesi (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
Similarity:
Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz , definita su un sottoinsieme di uno spazio di Hilbert a valori compatti e convessi in , può essere estesa su tutto ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz .
Jakub Duda (2009)
Fundamenta Mathematicae
Similarity:
We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of and functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes and (introduced by Preiss and Laczkovich) for Cⁿ smoothness....
Naotsugu Chinen (2015)
Commentationes Mathematicae Universitatis Carolinae
Similarity:
By , , we denote the -th symmetric product of a metric space as the space of the non-empty finite subsets of with at most elements endowed with the Hausdorff metric . In this paper we shall describe that every isometry from the -th symmetric product into itself is induced by some isometry from into itself, where is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence...
Mikołaj Krupski, Witold Marciszewski (2012)
Colloquium Mathematicae
Similarity:
We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into . We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into , but fails to embed...
Grzegorz Bartuzel, Andrzej Fryszkowski (2014)
Banach Center Publications
Similarity:
In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices are commutative and the multifunction is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that y₀ ∈ W4,1 a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*)...
Luděk Zajíček (2013)
Colloquium Mathematicae
Similarity:
We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...
Vanda Fülöp (2006)
Colloquium Mathematicae
Similarity:
Denote by the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.
Thomas Vils Pedersen (2004)
Studia Mathematica
Similarity:
For 0 < γ ≤ 1, let 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 be the closed ideal in consisting of those functions for which (i) f = 0 on E, (ii) as d(z,E),d(w,E) → 0, (iii) . Also, for a closed ideal I in , let = z ∈ : f(z) = 0 for every f ∈ I and let be the greatest common divisor of the inner parts of non-zero functions...
Wojciech Zygmunt (2016)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
Similarity:
In this note we shall prove that for a continuous function , where , the paratingent of at is a non-empty and compact set in if and only if satisfies Lipschitz condition in a neighbourhood of . Moreover, in this case the paratingent is a connected set.
Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)
Banach Center Publications
Similarity:
Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put . Consider the integral functional G defined on some non--type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...
Constantin Costara, Dušan Repovš (2010)
Studia Mathematica
Similarity:
We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either or for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...
Cung The Anh, Nguyen Duong Toan (2014)
Annales Polonici Mathematici
Similarity:
We prove the existence of uniform attractors in the space for the nonautonomous nonclassical diffusion equation , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors at ε = 0 is also studied.
Daniel Azagra, Mar Jiménez-Sevilla (2002)
Bulletin de la Société Mathématique de France
Similarity:
We study the size of the sets of gradients of bump functions on the Hilbert space , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space can be uniformly approximated by smooth Lipschitz functions so that the cones generated by the ranges of its derivatives have empty interior. This implies that there are smooth...
Tuomas Hytönen, Assaf Naor (2013)
Studia Mathematica
Similarity:
Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and and are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by , we show that for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...
J. Marshall Ash, Hajrudin Fejzić (2005)
Studia Mathematica
Similarity:
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. , 1 ≤ p ≤ ∞) sense at if there are numbers , |α| ≤ n, such that is in the approximate (resp. ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or 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...
B. Bojarski (2011)
Bulletin of the Polish Academy of Sciences. Mathematics
Similarity:
For a function the notion of p-mean variation of order 1, is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space in terms of is directly related to the characterisation of by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.
Abraham Racca, Emmanuel Cabral (2016)
Mathematica Bohemica
Similarity:
Equiintegrability in a compact interval may be defined as a uniform integrability property that involves both the integrand and the corresponding primitive . The pointwise convergence of the integrands to some and the equiintegrability of the functions together imply that is also integrable with primitive and that the primitives converge uniformly to . In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...
Leokadia Białas-Cież (2011)
Banach Center Publications
Similarity:
Let E be a compact set in the complex plane, be the Green function of the unbounded component of with pole at infinity and where the supremum is taken over all polynomials of degree at most n, and . The paper deals with recent results concerning a connection between the smoothness of (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence . Some additional conditions are given for special classes of sets.
Stanisław Szufla (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
Similarity:
For the integral equation (1) below we prove the existence on an interval of a solution with values in a Banach space , belonging to the class , . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.
Christina Brech, Piotr Koszmider (2014)
Fundamenta Mathematicae
Similarity:
This paper is concerned with the isomorphic structure of the Banach space and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that does not have an orthogonal -decomposition, that is, it is not of the form for any Banach space X. The main local result is that it is consistent that does not embed isomorphically into , where is the cardinality of the continuum,...
Mordechay B. Levin, Meir Smorodinsky (2005)
Colloquium Mathematicae
Similarity:
We extend Champernowne’s construction of normal numbers to base b to the case and obtain an explicit construction of a generic point of the shift transformation of the set .
Dušan Pokorný, Luděk Zajíček (2022)
Czechoslovak Mathematical Journal
Similarity:
We give a complete characterization of closed sets whose distance function is DC (i.e., is the difference of two convex functions on ). Using this characterization, a number of properties of such sets is proved.
Saard Youyen, Suthep Suantai (2008)
Banach Center Publications
Similarity:
In this paper, we define the direct sum of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that has the H-property if and only if each has the H-property, and has the Schur property if and only if each has the Schur property. Moreover, we also show that is rotund if and only if each is rotund.
Petr Petráček, Jiří Spurný (2016)
Commentationes Mathematicae Universitatis Carolinae
Similarity:
We provide a complex version of a theorem due to Bednar and Lacey characterizing real -preduals. Hence we prove a characterization of complex -preduals via a complex barycentric mapping.