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On a decomposition of Banach spaces

100%

Duda J.

Colloquium Mathematicum

2007

tom 108

nr 1

147-157

By using D. Preiss' approach to a construction from a paper by J. Matoušek and E. Matoušková, and some results of E. Matoušková, we prove that we can decompose a separable Banach space with modulus of convexity of power type p as a union of a ball small set (in a rather strong symmetric sense) and a set which is Aronszajn null. This improves an earlier unpublished result of E. Matoušková. As a corollary, in each separable Banach space with modulus of convexity of power type p, there exists a closed nonempty set A and a Borel non-Haar null set Q such that no point from Q has a nearest point in A. Another corollary is that ℓ₁ and L₁ can be decomposed as unions of a ball small set and an Aronszajn null set.

Generalized α-variation and Lebesgue equivalence to differentiable functions

100%

Duda J.

Fundamenta Mathematicae

2009

tom 205

nr 3

191-217

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 $CBVG_{1/n}$ and $SBVG_{1/n}$ 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 $CBV_{1/n}$ and $SBV_{1/n}$ (introduced by Preiss and Laczkovich) for Cⁿ smoothness. As a consequence, we deduce that Lebesgue equivalence to an n-times differentiable function is the same as Lebesgue equivalence to a function f which is (n-1)-times differentiable with $f^{(n-1)}(·)$ pointwise Lipschitz. We also characterize functions that are Lebesgue equivalent to n-times differentiable functions with a.e. nonzero derivatives. As a corollary, we establish a generalization of Zahorski's Lemma for higher order differentiability.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

2 Duda J.

1 2009

1 2007

Wyniki wyszukiwania

On a decomposition of Banach spaces

Generalized α-variation and Lebesgue equivalence to differentiable functions