*As an application of this abstract theorem, we will show that every lattice homomorphism T:W _{0[1,p](Î©_1) -->W0[1,q](Î©_2) has the form T(u)(y) = u(Î¾(y))g(y) quasi everywhere on Î©_2 where Î¾:Î©_2 -->Î©_1 and g:Î©_2 --> [0, âˆž). }*