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Positivity

, Volume 14, Issue 4, pp 637–653 | Cite as

Function spaces arising from kernel operators

  • Guillermo P. CurberaEmail author
  • Werner J. Ricker
Article
  • 54 Downloads

Abstract

Given a probability space (Ω, μ) and a rearrangement invariant space X on [0,1], in certain situations inequalities for spaces of \({\mathbb {R}}\)-valued functions on Ω are equivalent to the boundedness of an associated operator T K : L ([0, 1]) → X generated by a kernel K ≥ 0 on the unit square (e.g. Sobolev type inequalities or Riesz potentials on subsets \({\Omega \subset \mathbb {R}^n}\)). A natural class of spaces for treating such inequalities is given by \({[T_{K}, X](\Omega) := \{u : \Omega\to \mathbb {R} : T_{K} u^* \in X\}}\) together with the functional \({u \mapsto ||T_{K} u^*||_X}\), where u* is the decreasing rearrangement of u. The investigation of these spaces is our main aim; the nature of the base space X and of K (via its monotonicity/growth properties) play a crucial role.

Keywords

Optimal domain Kernel operator Rearrangement invariant spaces Quasi-normed spaces 

Mathematics Subject Classification (2000)

Primary 46E30 47G10 Secondary 46A40 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Math.–Geogr. FakultätKatholische Universität Eichstätt–IngolstadtEichstättGermany

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