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A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces

Katrin Schumacher (2009)

Czechoslovak Mathematical Journal

Given a domain Ω of class C k , 1 , k , we construct a chart that maps normals to the boundary of the half space to normals to the boundary of Ω in the sense that ( - x n ) α ( x ' , 0 ) = - N ( x ' ) and that still is of class C k , 1 . As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to k on domains of class C k , 1 . The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.

A C(K) Banach space which does not have the Schroeder-Bernstein property

Piotr Koszmider (2012)

Studia Mathematica

We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...

A class of Banach sequence spaces analogous to the space of Popov

Parviz Azimi, A. A. Ledari (2009)

Czechoslovak Mathematical Journal

Hagler and the first named author introduced a class of hereditarily l 1 Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily l p Banach spaces for 1 p < . Here we use these spaces to introduce a new class of hereditarily l p ( c 0 ) Banach spaces analogous of the space of Popov. In particular, for p = 1 the spaces are further examples of hereditarily l 1 Banach spaces failing the Schur property.

A class of weighted convolution Fréchet algebras

Thomas Vils Pedersen (2010)

Banach Center Publications

For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that ω m ( t ) / ω ( t ) as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...

A commutator theorem with applications.

Mario Milman (1993)

Collectanea Mathematica

We give an extension of the commutator theorems of Jawerth, Rochberg and Weiss [9] for the real method of interpolation. The results are motivated by recent work by Iwaniek and Sbordone [6] on generalized Hodge decompositions. The main estimates of these authors are based on a commutator theorem for a specific operator acting on Lp spaces and through the use of the complex method of interpolation. In this note we give an extension of the Iwaniek-Sbordone theorem to general real interpolation scales....

A compact set without Markov’s property but with an extension operator for C -functions

Alexander Goncharov (1996)

Studia Mathematica

We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator L : ( K ) C [ 0 , 1 ] . At the same time, Markov’s inequality is not satisfied for certain polynomials on K.

A connection between multiplication in C(X) and the dimension of X

Andrzej Komisarski (2006)

Fundamenta Mathematicae

Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.

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