Integrability, mean convergence, and parseval's formula for double trigonometric series

Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order (p, 0), (0, p), and (p, p) with the weight (|j|̄ |k|̄)^p-1 for some p > 1. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted L^r-integrability and weighted L^r-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], Móricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].