# Function spaces arising from kernel operators

Function spaces arising from kernel operators 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Function spaces arising from kernel operators

, Volume 14 (4) – Jun 15, 2010
17 pages

/lp/springer_journal/function-spaces-arising-from-kernel-operators-pkCJAfyeyC
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0069-5
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Jun 15, 2010

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