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Kernel theorems in spaces of generalized functions

Antoine Delcroix (2010)

Banach Center Publications

In analogy to the classical isomorphism between ((ℝⁿ), ' ( ℝ m ) ) and ' ( ℝ m + n ) (resp. ( ( ℝ ⁿ ) , ' ( ℝ m ) ) and ' ( ℝ m + n ) ), we show that a large class of moderate linear mappings acting between the space C ( ℝ ⁿ ) of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space ( ℝ ⁿ ) of Colombeau rapidly decreasing generalized functions and the space τ ( ℝ ⁿ ) of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of ( ℝ m + n ) (resp. τ ( ℝ m + n ) ). The main novelty is to use accelerated...

Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces

F. A. Sukochev, D. Zanin (2009)

Studia Mathematica

We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which ∥ ∑ k = 1 n ξ k ∥ E ≤ C n q , where ξ k k ≥ 1 ⊂ E is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces e x p ( L p ) , p ≥ 1. We further apply our results to the study of Banach-Saks index sets in...

K-metric and K-normed linear spaces: survey.

P. P. Zabrejko (1997)

Collectanea Mathematica

We give a short survey on some fixed point theorems which are generalizations of the classical Banach-Caccioppoli principle of contractive mappings. All these results are gathered in three theorems about existence and uniqueness of fixed points for operators which act in K-metric or K-normed linear spaces and, in particular, in local convex spaces and scales of Banach spaces. Three fixed point theorems presented in this article cover numerous applications in numerical methods, theory of integral...

Kneser-type theorem for the Darboux problem in Banach spaces

Mieczysław Cichoń, Ireneusz Kubiaczyk (2001)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the Darboux problem in some class of Banach spaces. The right-hand side of this problem is a Pettis-integrable function satisfying some conditions expressed in terms of measures of weak noncompactness. We prove that the set of all local pseudo-solutions of our problem is nonempty, compact and connected in the space of continuous functions equipped with the weak topology.

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