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1
Content available remote

Corrigenda to: "Optimal domains for the kernel operator associated with Sobolev's inequality" (Studia Math. 158 (2003), 131-152)

100%
Curbera G., Ricker W.
Studia Mathematica
|
2005
|
tom 170
|
nr 2
217-218
2
Content available remote

Optimal domains for the kernel operator associated with Sobolev's inequality

100%
Curbera G., Ricker W.
Studia Mathematica
|
2003
|
tom 158
|
nr 2
131-152
EN
Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on [0,1]. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its optimal domain. Several results are devoted to identifying the maximal rearrangement invariant space inside the optimal domain. The methods and techniques used involve interpolation theory, Banach function spaces and vector integration.
3
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Characterizing Fréchet-Schwartz spaces via power bounded operators

81%
Albanese A., Bonet J., Ricker W.
Studia Mathematica
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2014
|
tom 224
|
nr 1
25-45
EN
We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Fréchet-Montel spaces with a basis. Every strongly convergent sequence of continuous linear operators on a Fréchet-Schwartz space does so in a special way. We single out this type of "rapid convergence" for a sequence of operators and study its relationship to the structure of the underlying space. Its relevance for Schauder decompositions and the connection to mean ergodic operators on Fréchet-Schwartz spaces is also investigated.
4
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Operator ideal properties of vector measures with finite variation

81%
Okada S., Ricker W., Rodríguez-Piazza L.
Studia Mathematica
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2011
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tom 205
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nr 3
215-249
EN
Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the membership of Iₘ in various classical operator ideals(e.g., the compact, p-summing, completely continuous operators). Depending on which operator ideal is under consideration, the geometric nature of the Banach space X may also play a crucial role. Of particular importance in this regard is whether or not X contains an isomorphic copy of the classical sequence space ℓ¹. The compact range property of X is also relevant.
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