Lattice homomorphisms between Sobolev spaces
The author shows that every vector lattice homomorphism \(T\) from \(W^{1,p}_{0}(\Omega_{1})\) into \(W^{1,q}(\Omega_{2})\) has the representation \(Tu=(u \circ \xi)g\) (\(\text{Cap}_{q}\)-quasi everywhere on \(\Omega_{2}\)). Here, \(p,q\in (1,\infty)\), \(\Omega_{1}\) and \(\Omega_{2}\) are open sets in \(\mathbb R^{n}\), \(\xi\) maps \(\Omega_{1}\) to \(\Omega_{2}\), and \(g\) maps \(\Omega_{2}\) to \([0,\infty)\). This is a corollary of a more general and abstract representation theorem, which the author also proves. Some other applications of the theorem and examples are given as well.
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