On weighted integrability of double cosine series (Q1359593)

The authors define the class \(L(p,\alpha,\beta)\) of even, periodic functions \(f(x,y)\) by the requirement \[ |f|_{p,\alpha,\beta}:= \Biggl\{\int^\pi_0 \int^\pi_0|f(x,y)|^p(\sin x)^{\alpha p}(\sin y)^{\beta p}dx dy\Biggr\}^{1/p}< \infty. \] They prove lower or/and upper estimates for this norm in the special case when the Fourier series \(\sum^\infty_{m=1} \sum^\infty_{n=1} a_{mn}\cos mx\cos ny\) of \(f\) is such that some of the conditions \[ a_{mn}\geq 0,\quad\Delta_{10} a_{mn}\geq 0,\quad \Delta_{01}a_{mn}\geq 0,\quad\Delta_{11}a_{mn}\geq 0 \] are satisfied. In this way, they extend two theorems of \textit{R. Askey} and \textit{S. Wainger} [Duke Math. J. 33, 223-228 (1966; Zbl 0136.36501)] and a theorem of the reviewer [Proc. Am. Math. Soc. 109, No. 2, 417-425 (1990; Zbl 0741.42010)].