This is an announcement for the paper "Lattice homomorphisms between Sobolev spaces" by Markus Biegert.
Abstract: We show that every vector lattice homomorphism $T$ from $W^{1,p}_0(\Omega_1)$ into $W^{1,q}(\Omega_2)$ for $p,q\in (1,\infty)$ and open sets \Omega_1,\Omega_2\subset\IR^N$ has a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_q$-quasi everywhere on }\Omega_2$ with mappings $\xi:\Omega_2\to\Omega_1$ and $g:\Omega_2\to[0,\infty)$. This representation follows as an application of an abstract and more general representation theorem, which can be applied in many other situations. We prove that every lattice homomorphism $T$ from $\tsW^{1,p}(\Omega_1)$ into $W^{1,q}(\Omega_2)$ admits a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_q$-quasi everywhere on }\Omega_2$ with mappings $\xi:\Omega_2\to\overline\Omega_1$ and $g:\Omega_2\to[0,\infty)$. Here $\tsW^{1,p}(\Omega_1)$ denotes the closure of $W^{1,p}(\Omega_1)\cap C_c(\overline\Omega_1)$ in $W^{1,p}(\Omega_1)$ and every $u\in\tsW^{1,p}(\Omega_1)$ admits a trace on the boundary $\partial\Omega_1$ of $\Omega_1$. Finally we prove that every lattice homomorphism $T$ from $\tsW^{1,p}(\Omega_1)$ into $\tsW^{1,q}(\Omega_2)$ where $\Omega_1$ is bounded has a representation of the form $Tu=(u\circ\xi)g\quad\mbox{ $\Cap_{q,\Omega_2}$-quasi everywhere on }\overline\Omega_2$ with mappings $\xi:\overline\Omega_2\to\overline\Omega_1$ and $g:\overline\Omega_2\to[0,\infty)$. At the end of this article we consider also lattice isomorphisms between Sobolev spaces and the representation of their inverses.
Archive classification: math.AP math.FA
The source file(s), orderhomomorphism.bbl: 4468 bytes
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