We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued
function f ∈ L1 (ℝ2) with bounded support at a given point (x0, g0) ∈ ℝ2. It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier
integral of the marginal functions f(x, y0), x ∈ ℝ, and f(x0, y), y ∈ ℝ, at x:= x0 and y:= y0, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.