Commuting Dual Toeplitz Operators on the Harmonic Dirichlet Space

In this paper,we characterize the symbols for（semi-）commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space.We show that for φ,ψ∈W1,∞,SφSψ=Sψ Sφ on（Dh）⊥ if and only if φ and ψ satisfy one of the following conditions:（1） Both φ and ψ are harmonic functions;（2） There exist complex constants α and β,not both 0,such that φ=αψ +β.