Leafwise homotopy equivalences and leafwise Sobolov spaces

We prove that a leafwise homotopy equivalence between compact foliated manifolds induces a well defined bounded operator between all Sobolov spaces of leafwise (for the natural foliations of the graphs of the original foliations) differential forms with coefficients in a leafwise flat bundle. We further prove that the associated map on the leafwise reduced L 2 cohomology is an isomorphism which only depends on the leafwise homotopy class of the homotopy equivalence.