Extension maps in ultradifferentiable and ultraholomorphic function spaces

Jean Schmets; Manuel Valdivia

Displaying similar documents to “Extension maps in ultradifferentiable and ultraholomorphic function spaces”

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

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We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $Σ ⁰_{α}$ , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $Π ⁰_{α}$ uniformization which is the graph of a $Σ ⁰_{α}$ -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{δ}$ sections. ...

Effective decomposition of σ-continuous Borel functions

Gabriel Debs (2014)

Fundamenta Mathematicae

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We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ${(A ₙ)}_{n \in ω}$ of X such that $f_{| A ₙ}$ is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

Failure of the Factor Theorem for Borel pre-Hilbert spaces

Tadeusz Dobrowolski, Witold Marciszewski (2002)

Fundamenta Mathematicae

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In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{σ δ σ}$ -subset of X and contains a retract R so that $R \times E^{ω}$ is not homeomorphic to $E^{ω}$ . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

The effective Borel hierarchy

M. Vanden Boom (2007)

Fundamenta Mathematicae

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Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is $Σ_{α}$ (respectively, $Π_{α}$ ) in the Borel hierarchy iff K is axiomatized by an infinitary $Σ_{α}$ (respectively, $Π_{α}$ ) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective $Σ_{α}$ or effective $Π_{α}$ Borel set with a computable infinitary sentence of the same complexity....

On the complexity of subspaces of $S_{ω}$

Carlos Uzcátegui (2003)

Fundamenta Mathematicae

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Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^{X}$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{ω}$ is $F_{σ δ}$ . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{ω}$ . We show that $S_{ω}$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology....

$F_{σ}$ -mappings and the invariance of absolute Borel classes

Petr Holický, Jiří Spurný (2004)

Fundamenta Mathematicae

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It is proved that $F_{σ}$ -mappings preserve absolute Borel classes, which improves results of R. W. Hansell, J. E. Jayne and C. A. Rogers. The proof is based on the fact that any $F_{σ}$ -mapping f: X → Y of an absolute Suslin metric space X onto an absolute Suslin metric space Y becomes a piecewise perfect mapping when restricted to a suitable $F_{σ}$ -set $X_{\infty} \subset X$ satisfying $f (X_{\infty}) = Y$ .

Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara (2015)

Fundamenta Mathematicae

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Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{σ}$ set under that function is again $F_{σ}$ . Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

Banach spaces of bounded Szlenk index

E. Odell, Th. Schlumprecht, A. Zsák (2007)

Studia Mathematica

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For a countable ordinal α we denote by $_{α}$ the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each $_{α}$ admits a separable, reflexive universal space. We also show that spaces in the class $_{ω^{α \cdot ω}}$ embed into spaces of the same class with a basis. As a consequence we deduce that each $_{α}$ is analytic in the Effros-Borel structure of subspaces of C[0,1].

Equilibrium states for interval maps: the potential $- t log | D f |$

Henk Bruin, Mike Todd (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $f : I \to I$ be a $C^{2}$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $φ_{t} : x \mapsto - t log | D f (x) |$ for $t$ close to $1$ , and also that the pressure function $t \mapsto P (φ_{t})$ is analytic on an appropriate interval near $t = 1$ .

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$ -subset $A_{k}$ of X such that $d i m A_{k} \leq k$ and the restriction $f | (X ∖ A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

The mean square of the divisor function

Chaohua Jia, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that $\sum_{n \leq x} d ² (n) = x P (l o g x) + E (x)$ , where P(y) is a cubic polynomial in y and $E (x) = O (x^{3 / 5 + ε})$ , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E (x) = O (x^{1 / 2 + ε})$ . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E (x) = O (x^{1 / 2} (l o g x) ⁵ l o g l o g x)$ . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E (x) = O (x^{1 / 2} (l o g x) ⁵)$ . ...

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine, Katsuro Sakai, Masato Yaguchi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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By Fin(X) (resp. $F i n^{k} (X)$ ), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and $ℓ ₂^{f} (τ)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E = ℓ ₂^{f} (τ)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes $_{α} (τ)$ and $_{α} (τ)$ of weight ≤ τ (α > 0) then Fin(E) and each $F i n^{k} (E)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X)...

Preservation of properties of a map by forcing

Akira Iwasa (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $f : X \to Y$ be a continuous map such as an open map, a closed map or a quotient map. We study under what circumstances $f$ remains an open, closed or quotient map in forcing extensions.

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...

Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces

S. Moll (2008)

Banach Center Publications

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Let X be a completely regular Hausdorff topological space and $C_{p} (X)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_{p} (X)$ , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_{p} (X)$ .

Relations between Shy Sets and Sets of $ν_{p}$ -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a non-zero non-atomic translation-invariant Borel measure $ν_{p}$ on the Banach space $ℓ_{p} (1 \leq p \leq \infty)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " $ν_{p}$ -almost every element of $ℓ_{p}$ has a property P" implies that “almost every” element of $ℓ_{p}$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

Remarks on WDC sets

Dušan Pokorný, Luděk Zajíček (2021)

Commentationes Mathematicae Universitatis Carolinae

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We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A \subset ℝ^{2}$ . We prove that, for such $A$ , the distance function $d_{A} = dist (\cdot, A)$ is a “DC aura” for $A$ , which implies that each closed locally WDC set in $ℝ^{2}$ is a WDC set. Another consequence is that compact WDC subsets of $ℝ^{2}$ form a Borel subset of the space of all compact sets.

Some characterizations of the class $_{m} (Ω)$ and applications

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

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We give some characterizations of the class $_{m} (Ω)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

Gebelein's inequality and its consequences

M. Beśka, Z. Ciesielski (2006)

Banach Center Publications

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Let $(X_{i}, i = 1, 2, . . .)$ be the normalized gaussian system such that $X_{i} \in N (0, 1)$ , i = 1,2,... and let the correlation matrix $ρ_{i j} = E (X_{i} X_{j})$ satisfy the following hypothesis: $C = s u p_{i \geq 1} \sum_{j = 1}^{\infty} | ρ_{i, j} | < \infty$ . We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces

Boban Karapetrović (2018)

Czechoslovak Mathematical Journal

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We show that if $α > 1$ , then the logarithmically weighted Bergman space $A_{{log}^{α}}^{2}$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{α - 1}}^{2}$ , while if $α > 2$ and $0 < ε \leq α - 2$ , then the Hilbert matrix operator $H$ maps $A_{{log}^{α}}^{2}$ into $A_{{log}^{α - 2 - ε}}^{2}$ .We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space $ℬ_{{log}^{α}}$ , $α \in ℝ$ , into itself, while $H$ maps $ℬ_{{log}^{α}}$ into $ℬ_{{log}^{α + 1}}$ .In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space $ℬ_{{log}^{α}}^{1}$ , $α > 0$ , into $ℬ_{{log}^{α - 1}}^{1}$ . We show that this result is sharp. We also show that $H$ maps $ℬ_{{log}^{α}}^{1}$ , $α \geq 0$ ,...

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila, Mikhail Lyubich, Weixiao Shen (2011)

Journal of the European Mathematical Society

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We study the parameter space of unicritical polynomials $f_{c} : z \mapsto z^{d} + c$ . For complex parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

A proof of the crossing number of $K_{3, n}$ in a surface

Pak Tung Ho (2007)

Discussiones Mathematicae Graph Theory

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In this note we give a simple proof of a result of Richter and Siran by basic counting method, which says that the crossing number of $K_{3, n}$ in a surface with Euler genus ε is ⎣n/(2ε+2)⎦ n - (ε+1)(1+⎣n/(2ε+2)⎦).

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{α}$ , $Y_{α}$ and $Z_{α}$ such that (i) $f X_{α}, f Y_{α}, f Z_{α} = ω ₀$ , where f is either trdef or ₀-trsur, (ii) $A (α) - t r i n d X_{α} = \infty$ and $M (α) - t r i n d X_{α} = - 1$ , (iii) $A (α) - t r i n d Y_{α} = - 1$ and $M (α) - t r i n d Y_{α} = \infty$ , and (iv) $A (α) - t r i n d Z_{α} = M (α) - t r i n d Z_{α} = \infty$ and $A (α + 1) \cap M (α + 1) - t r i n d Z_{α} = - 1$ . We also show that there exists no separable metrizable space $W_{α}$ with $A (α) - t r i n d W_{α} \neq \infty$ , $M (α) - t r i n d W_{α} \neq \infty$ and $A (α) \cap M (α) - t r i n d W_{α} = \infty$ , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

Remarks on Ramanujan's inequality concerning the prime counting function

Mehdi Hassani (2021)

Communications in Mathematics

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In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that $π {(x)}^{2} < \frac{e x}{log x} π (\frac{x}{e})$ for $x$ sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor $\frac{x}{log x}$ on its right hand side by the factor $\frac{x}{log x - h}$ for a given $h$ , and by replacing the numerical factor $e$ by a given positive $α$ . Finally, we introduce and study inequalities...

< $> 𝒜 <$ >-Schemes and Zariski-Riemann Spaces

Satoshi Takagi (2012)

Rendiconti del Seminario Matematico della Università di Padova

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A classification of inverse limit spaces of tent maps with periodic critical points

Lois Kailhofer (2003)

Fundamenta Mathematicae

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We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps $f_{a}$ , $f_{b}$ with periodic critical points, we show that the inverse limit spaces $(_{a}, f_{a})$ and $(_{b}, g_{b})$ are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

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For the real quadratic map $P_{a} (x) = x^{2} + a$ and a given $ϵ > 0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_{a}^{n} |_{J}$ univalent, with bounded distortion and $B (0, ϵ) \subseteq P_{a}^{n} (J)$ for some $n \in ℕ$ . The $ϵ$ -weakly expanding set is the set of points which do not have good expansion properties. Let $α$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[α, - α]$ . We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

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If $(M, \nabla)$ is a manifold with a symmetric linear connection, then $T^{*} M$ can be endowed with the natural Riemann extension $\bar{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\bar{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*} M, \bar{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\bar{g}$ reduces...