# Lattice homomorphisms between Sobolev spaces

Lattice homomorphisms between Sobolev spaces We show in Theorem 4.4 that every vector lattice homomorphism T from $${\mathsf{W}^{1,p}_0(\Omega_1)}$$ into $${\mathsf{W}^{1,q}(\Omega_2)}$$ for $${p,q\in (1,\infty)}$$ and open sets $${\Omega_1,\Omega_2\subset\mathbb{R}^N}$$ has a representation of the form $${T\mathsf{u}=(\mathsf{u}\circ\xi)g}$$ (Cap q -quasi everywhere on Ω2) with mappings ξ : Ω2 → Ω1 and g : Ω2 → [0, ∞). This representation follows as an application of an abstract and more general representation theorem (Theorem 3.5). Other applications are also given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Lattice homomorphisms between Sobolev spaces

, Volume 14 (2) – Jun 16, 2009
19 pages

/lp/springer_journal/lattice-homomorphisms-between-sobolev-spaces-R9OVvx0rr8
Publisher
Springer Journals
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/s11117-009-0022-7
Publisher site
See Article on Publisher Site

### Abstract

We show in Theorem 4.4 that every vector lattice homomorphism T from $${\mathsf{W}^{1,p}_0(\Omega_1)}$$ into $${\mathsf{W}^{1,q}(\Omega_2)}$$ for $${p,q\in (1,\infty)}$$ and open sets $${\Omega_1,\Omega_2\subset\mathbb{R}^N}$$ has a representation of the form $${T\mathsf{u}=(\mathsf{u}\circ\xi)g}$$ (Cap q -quasi everywhere on Ω2) with mappings ξ : Ω2 → Ω1 and g : Ω2 → [0, ∞). This representation follows as an application of an abstract and more general representation theorem (Theorem 3.5). Other applications are also given.

### Journal

PositivitySpringer Journals

Published: Jun 16, 2009

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