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,qin (1,infty)}) and open sets ({Omega_1,Omega_2subsetmathbb{R}^N}) has a representation of the form ({Tmathsf{u}=(mathsf{u}circxi)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.