We prove symmetric properties of the p-integrable subspaces $$T_p$$ T p of the universal Teichmüller space T, for $$p>0.$$ p > 0 . The elements of $$T_p$$ T p are quasisymmetric automorphisms of the real line or of the unit circle, which have a q.c. extension with p-integrable complex dilatation with respect to the Poincaré metric. We show that $$T_p\subset T_S,$$ T p ⊂ T S , where $$T_S$$ T S is the symmetric Teichmüller space and that, in the case of the real line, $$T_p\subset T_0,$$ T p ⊂ T 0 , where $$T_0$$ T 0 is the little Teichmüller space. The proofs use properties of the boundary behavior of q.c maps such as uniform weak conformality.
Analysis and Mathematical Physics – Springer Journals
Published: Oct 31, 2018