Sobolev space
Similarly it extends to a unitary on each Sobolev space H"k"(T2) with the same property. It is the counterpart on the torus of the Beurling transform. The standard theory of Fredholm operators shows that the operators corresponding to "I" – "μ" "U" and "I" – "U" "μ" are invertible on each Sobolev space. On the other hand,