Displaying similar documents to “Borel summable solutions of the Burgers equation”

Power means and the reverse Hölder inequality

Victor D. Didenko, Anatolii A. Korenovskyi (2011)

Studia Mathematica

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Let w be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse Hölder inequality with exponents 0 < α < β. In the present paper, sharp estimates of the compositions of the power means α w ( x ) : = ( ( 1 / x ) 0 x w α ( t ) d t ) 1 / α , x > 0, are obtained for various exponents α. As a result, for the function w a property of self-improvement of summability exponents is established.

Borel sets with σ-compact sections for nonseparable spaces

Petr Holický (2008)

Fundamenta Mathematicae

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We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections E x = y Y : ( x , y ) E , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.

Singular non polynomial perturbations of - Δ + | x | 2

Franco Nardini (1983)

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

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Si studia la perturbazione dello spettro deiroperatore - Δ + | x | 2 dovuta all'introduzione di un potenziale singolare non polinomiale e si prova che la serie perturbativa del primo autovalore di tale operatore è sommabile secondo Borel.

Summability of first integrals of a C ω -non-integrable resonant Hamiltonian system

Masafumi Yoshino (2012)

Banach Center Publications

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This article studies the summability of first integrals of a C ω -non-integrable resonant Hamiltonian system. The first integrals are expressed in terms of formal exponential transseries and their Borel sums. Smooth Liouville integrability and a relation to the Birkhoff transformation are discussed from the point of view of the summability.

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 α , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a Π α uniformization which is the graph of a Σ 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 ω of X such that f | A is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space κ κ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in κ κ for uncountable regular κ is however consistent (with...

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....

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....

Pointwise limit theorem for a class of unbounded operators in r -spaces

Ryszard Jajte (2007)

Studia Mathematica

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We distinguish a class of unbounded operators in r , r ≥ 1, related to the self-adjoint operators in ². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin’s criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in r -spaces are applied.

On generalized absolute Cesaro summability factors

A. Nihal Tuncer (2002)

Annales Polonici Mathematici

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Using δ-quasi-monotone and any almost increasing sequences we prove a theorem on | C , α , β ; δ | k summability factors of infinite series, which generalizes a theorem of Mazhar [7] on | C , 1 | k summability factors.

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 X satisfying f ( X ) = 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...

On convergence sets of divergent power series

Buma L. Fridman, Daowei Ma, Tejinder S. Neelon (2012)

Annales Polonici Mathematici

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A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family y = φ s ( t , x ) = s b ( x ) t + b ( x ) t ² + of analytic curves in ℂ × ℂⁿ passing through the origin, C o n v φ ( f ) of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series f ( φ s ( t , x ) , t , x ) converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that E = C o n v φ ( f ) if...

Borel spaces

K. P. S. Bhaskara Rao, B. V. Rao

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CONTENTSIntroduction............................................................................... 5Chapter 1. Borel spaces........................................................ 7 § 1. Borel spaces....................................................... 7 § 2. Classical descriptive set theory............................... 10 § 3. Measure and category............................................... 12 § 4. Countably generated structures.............................. 13 § 5. Product...

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 · 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].

Multi-dimensional Fejér summability and local Hardy spaces

Ferenc Weisz (2009)

Studia Mathematica

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It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space W ( h p , ) to W ( L p , ) . This implies the almost everywhere convergence of the Fejér means in a cone for all f W ( L , ) , which is larger than L ( d ) .

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 × E ω is not homeomorphic to E ω . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

On the Nörlund means of Vilenkin-Fourier series

István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)

Czechoslovak Mathematical Journal

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We prove and discuss some new ( H p , L p ) -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients { q k : k 0 } . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of...