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1
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Biseparating maps on generalized Lipschitz spaces

100%
Leung D.
Studia Mathematica
|
2010
|
tom 196
|
nr 1
23-40
EN
Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F-valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are characterized as weighted composition operators, i.e., of the form $Tf(y) = S_{y}(f(h^{-1}(y)))$ for a family of vector space isomorphisms $S_{y}: E → F$ and a homeomorphism h: X → Y. We also investigate the continuity of T and related questions. Here the functions involved (as well as the metric spaces X and Y) may be unbounded. Also, the arguments do not require the use of compactification of the spaces X and Y.
2
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ℓ¹-Spreading models in subspaces of mixed Tsirelson spaces

64%
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.
3
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Extension of functions with small oscillation

64%
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.
4
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Order isomorphisms on function spaces

64%
Leung D., Li L.
Studia Mathematica
|
2013
|
tom 219
|
nr 2
123-138
EN
The classical theorems of Banach and Stone (1932, 1937), Gelfand and Kolmogorov (1939) and Kaplansky (1947) show that a compact Hausdorff space X is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of the space C(X). In this paper, it is shown that for rather general subspaces A(X) and A(Y) of C(X) and C(Y), respectively, any linear bijection T: A(X) → A(Y) such that f ≥ 0 if and only if Tf ≥ 0 gives rise to a homeomorphism h: X → Y with which T can be represented as a weighted composition operator. The three classical results mentioned above can be derived as corollaries. Generalizations to noncompact spaces and other function spaces such as spaces of Lipschitz functions and differentiable functions are presented.
5
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Functions of Baire class one

64%
Leung D., Tang W.-K.
Fundamenta Mathematicae
|
2003
|
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.
6
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A gauge approach to an ordinal index of Baire one functions

51%
Leung D., Tang W.-K., Zulijanto A.
Fundamenta Mathematicae
|
2010
|
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.
7
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Minimality properties of Tsirelson type spaces

51%
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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