Bounds for norms of sums of double trigonometric series with multiply monotone coefficients (Q1842470)

The authors consider double cosine, double sine, and cosine-sine series with special coefficients \(\{a_{mn} : m,n = 0,1, \dots\}\). Given nonnegative integers \(k_ 1\) and \(k_ 2\), the differences \(\Delta_{k_ 1 k_ 2 a_{mn}}\) are defined in the usual way. As it is well-known, if \(a_{mn} \to 0\) as \(m + n \to \infty\) and \(\Delta_{k_ 1 k_ 2 a_{mn}} \geq 0\) for all \(m,n\) and some \(k_ 1 \geq 1\), \(k_ 2 \geq 1\), then the double cosine, \(\dots\) series converge in Pringsheim's sense, except possibly at \(x = 0\) or \(y = 0 \pmod {2 \pi}\). The reviewer [Proc. Am. Math. Soc. 109, No. 2, 417-425 (1990; Zbl 0741.42010)] proved upper estimates for the \(L^ p\)-norm of the maximum of the rectangular partial sums, and lower estimates for the \(L^ p\)- norm of the sum of double cosine and double sine series, in the cases \(k_ 1 = k_ 2 = 1\) and \(1 \leq p < \infty\). Now, the present authors extend these estimates for the cases \(k_ 1\), \(k_ 2 = 1\) or 2, respectively, \(0 < p < \infty\), and for cosine-sine series, too, but without considering the maximal function.