Estimates for mixed norms of the sums of double trigonometric series with multiply monotonous coefficients (Q1390442)

We recall that a function \(f(x,y)\) is said to belong to the space \(L_{p_1,p_2}(- \pi,\pi)^2\) for some \(0< p_1, p_2< \infty\) if \(f(x, y)\) is \(2\pi\)-periodic with respect to each variable, measurable on \((-\pi, \pi)\), and such that \[ \| f\|_{p_1,p_2}:= \Biggl\{ \int^\pi_{-\pi} \Biggl(\int^\pi_{-\pi}| f(x,y)|^{p_1} dx\Biggr)^{p_2/p_1} dy\Biggr\}^{1/p_2}< \infty. \] The authors prove various estimates in \(L_{p_1,p_2}\)-norm of the sums of double trigonometric series in sines and/or cosines with multiply monotonous coefficients. Reviewer's remark: The reviewer was one of those who initiated the estimation of such sums in \(L_p(-\pi, \pi)^2\)-norm, \(1\leq p<\infty\) [see, for example, Proc. Am. Math. Soc. 102, No. 3, 633-640 (1988; Zbl 0666.42004); or Proc. Am. Math. Soc. 109, No. 2, 417-425 (1990; Zbl 0741.42010)].