<a name="abs1"/>Abstract??We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued functionf?L1(?2) with bounded support at a given point (x0,y0) ? ?2. It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functionsf(x,y0),x? ?, andf(x0,y),y? ?, at the pointsx:=x0andy:=y0, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.