Toeplitz and Hankel operators with symbols on Dirichlet space

We show that Toeplitz or small Hankel operators with symbols in L ∞ , 1 is a generalization of the case with the harmonic symbol in C ⊕ D ⊕ D ¯ , where D is the Dirichlet space on D . We also give a decomposition theorem for the Sobolev space of first order on D . Using this result, some characterizations for algebraic properties of Toeplitz or small Hankel operators with symbols in L ∞ , 1 are given.