$P^λ$-spaces and $L^λ$-spaces

Nguyen To Nhu

Displaying similar documents to “ $P^{λ}$ -spaces and $L^{λ}$ -spaces”

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 $L i p (α ₁, . . ., α_{N})$ for some $0 < α ₁, . . ., α_{N} < 1$ and its marginal functions satisfy $f (\cdot, x ₂, . . ., x_{N}) \in L i p β ₁, . . ., f (x ₁, . . ., x_{N - 1}, \cdot) \in L i p β_{N}$ for some $0 < β ₁, . . ., β_{N} < 1$ uniformly in the indicated variables $x_{l}$ , 1 ≤ l ≤ N, then $f ̃^{(η ₁, . . ., η_{N})} \in L i p (α ₁, . . ., α_{N})$ for each choice of $(η ₁, . . ., η_{N})$ with $η_{l} = 0$ or 1 for 1 ≤ l ≤ N.

A Sufficient Condition for the $C^{2}$ -Rectifiability of the Set of Regular Values (in the Sense of Clarke) of a Lipschitz Map

Silvano Delladio (2008)

Bollettino dell'Unione Matematica Italiana

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We prove a result about the rectifiability of class $C^{2}$ of the set of regular values (in the sense of Clarke) of a Lipschitz map $\varphi\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$ with $n<N$

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz $M$ , definita su un sottoinsieme di uno spazio di Hilbert $H$ a valori compatti e convessi in $\mathbb{R}^{n}$ , può essere estesa su tutto $H$ ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz $M$ .

Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec (2009)

Studia Mathematica

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It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $C F L (_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $L I P (_{r})$ of equivalence classes of all real-valued Lipschitz maps on $_{r}$ . The space of all signed (real-valued) Borel measures on $_{r}$ is isometrically embedded in the dual space of $C F L (_{r})$ and it is shown that the image...

On Banach spaces C(K) isomorphic to c₀(Γ)

Witold Marciszewski (2003)

Studia Mathematica

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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 $ω_{ω}$ and with the third derived set $K^{(3)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c ₀ (ω_{ω})$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

Generalized α-variation and Lebesgue equivalence to differentiable functions

Jakub Duda (2009)

Fundamenta Mathematicae

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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 $C B V G_{1 / n}$ and $S B V G_{1 / n}$ 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 $C B V_{1 / n}$ and $S B V_{1 / n}$ (introduced by Preiss and Laczkovich) for Cⁿ smoothness....

Lipschitz equivalence of graph-directed fractals

Ying Xiong, Lifeng Xi (2009)

Studia Mathematica

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This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if ${E_{i}}_{i}$ and ${F_{j}}_{j}$ are dust-like graph-directed sets satisfying the transitivity condition, then $E_{i ₁}$ and $E_{i ₂}$ are Lipschitz equivalent, and $E_{i}$ and $F_{j}$ are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.

The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

Matematički Vesnik

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The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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On C $^{*}$ -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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On a characterization of $L^{p}$ -norm

Janusz Matkowski (1989)

Annales Polonici Mathematici

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$ℵ_{1}$ -incompactness of Z

Ralph McKenzie (1971)

Colloquium Mathematicae

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On the $p^{λ}$ problem

Stephan Baier (2004)

Acta Arithmetica

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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Lipschitz and uniform embeddings into $ℓ_{\infty}$

N. J. Kalton (2011)

Fundamenta Mathematicae

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We show that there is no uniformly continuous selection of the quotient map $Q : ℓ_{\infty} \to ℓ_{\infty} / c ₀$ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from $B_{X * *}$ onto $B_{X}$ .