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 ∈ W ^1,∞, S _φ S _ψ = S _ψ S _φ on (D _h )^⊥ 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.