Abstract. A theorem of Ferenc Lukács determines the jumps of a periodic, Lebesgue integrable function $f$ in terms of the partial sum of the conjugate series to the Fourier series of $f$. The aim of this paper is to prove an analogous theorem in terms of the Abel--Poisson mean for a periodic, Lebesgue integrable function in two variables. We also prove an estimate of the mixed partial derivative of the Abel--Poisson mean of the conjugate series to the Fourier series of an integrable function $F(x,y)$ at such a point, where $F$ is smooth. The two results are closely related.

AMS Subject Classification (1991): 42B05, 42A16

Keyword(s): Fourier series, conjugate series, rectangular partial sum, Abel--Poisson mean, generalized jump, smoothness of a function in two variables, \lambda_* ({\msbm T}^2), \Lambda_*({\msbm T}^2), Zygmund classesand

Received April 15, 2002, and in revised form August 14, 2002. (Registered under 2925/2009.)