Toeplitz and Hankel operators with L∞,1 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 , 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.