Injectivity of the quotient Bers embedding of Teichm\"uller spaces

The Bers embedding of theTeichm\"uller space is a homeomorphism into the Banach space of certain holomorphic automorphic forms. For a subspace of the universal Teichm\"uller space and its corresponding Banach subspace, we consider whether the Bers embedding can project down between their quotient spaces. If this is the case, it is called the quotient Bers embedding. Injectivity of the quotient Bers embedding is the main problem in this paper. Alternatively, we can describe this situation as the universal Teichm\"uller space having an affine foliated structure induced by this subspace. We give several examples of subspaces for which the injectivity holds true, including the Teichm\"uller space of circle diffeomorphisms with H\"older continuous derivative. As an application, the regularity of conjugation between representations of a Fuchsian group into the group of circle diffeomorphisms is investigated.