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1
Content available remote

ℓ¹-Spreading models in subspaces of mixed Tsirelson spaces

100%
Leung D., Tang W.-K.
Studia Mathematica
|
2006
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tom 172
|
nr 1
47-68
EN
We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X = T[(θₙ,𝓢ₙ)^{∞}_{n=1}]$: (1) Every block subspace of X contains an $ℓ¹-𝓢_{ω}$-spreading model, (2) The Bourgain ℓ¹-index $I_{b}(Y) = I(Y) > ω^{ω}$ for any block subspace Y of X, (3) $limₘ lim supₙ θ_{m+n}/θₙ > 0$ and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X.Moreover, if one (and hence all) of these conditions holds, then X is arbitrarily distortable.
2
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Extension of functions with small oscillation

100%
Leung D., Tang W.-K.
Fundamenta Mathematicae
|
2006
|
tom 192
|
nr 2
183-193
EN
A classical theorem of Kuratowski says that every Baire one function on a $G_{δ}$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that $β_{Y}(f) < ω^{α}$, α < ω₁, then f has an extension F to X so that $β_{X}(F) ≤ ω^{α}$. We also show that if f is a continuous real-valued function on Y, then f has an extension F to X so that $β_{X}(F) ≤ 3.$ An example is constructed to show that this result is optimal.
3
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Functions of Baire class one

100%
Leung D., Tang W.-K.
Fundamenta Mathematicae
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2003
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tom 179
|
nr 3
225-247
EN
Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β(f) ≤ ω^{ξ₁} · ω^{ξ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions converging to f pointwise such that $supₙβ(fₙ) ≤ ω^{ξ₁}$ and $γ((fₙ)) ≤ ω^{ξ₂}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) ≤ ω^{ξ₁}$ and $β(g) ≤ ω^{ξ₂}$, then $β(fg) ≤ ω^{ξ}$, where ξ = max{ξ₁+ξ₂,ξ₂+ξ₁}. These results do not assume the boundedness of the functions involved.
4
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A gauge approach to an ordinal index of Baire one functions

81%
Leung D., Tang W.-K., Zulijanto A.
Fundamenta Mathematicae
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2010
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tom 210
|
nr 2
99-109
EN
We develop a calculus for the oscillation index of Baire one functions using gauges analogous to the modulus of continuity.
5
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Minimality properties of Tsirelson type spaces

81%
Kutzarova D., Leung D., Manoussakis A., Tang W.-K.
Studia Mathematica
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2008
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tom 187
|
nr 3
233-263
EN
We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k})$, there is a further block basis $(y_{k})$ of $(x_{k})$ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k})$. Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain's ℓ¹-index are established. It is also shown that a large class of mixed Tsirelson spaces fails to be subsequentially minimal in a strong sense.
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