We consider the Duhamel equation φ ⊛ f = in the subspace C ∞ xy = f ∈ C ∞ ([0, 1] × [0, 1]) : f x, y = F xy for some F ∈ C ∞ [0, 1] of the space C ∞ ([0, 1] × [0, 1]) and prove that if φ xy=0 0, then this equation is uniquely solvable in C ∞ xy. The commutant of the restricted double integration operator W xy f xy := x 0 y 0 f (tτ) dτdt on C ∞ xy is also described. Some other related questions are also discussed.