Weighted integrability of double trigonometric series

Abstract

We study the double trigonometric series whose coefficients c j k c_{jk} are such that ∑ j = − ∞ ∞ ∑ k = − ∞ ∞ | c j k | > ∞ . \sum _{j=-\infty }^\infty \sum _{k=-\infty }^\infty |c_{jk}|>\infty . Then its rectangular partial sums converge uniformly to some f ∈ C ( T 2 ) f\in C(T^2) . We give sufficient conditions for the Lebesgue integrability of { f ( x , y ) − f ( x , 0 ) − f ( 0 , y ) + f ( 0 , 0 ) } ϕ ( x , y ) \{f(x,y)-f(x,0)-f(0,y)+f(0,0)\}\phi (x,y) , where ϕ ( x , y ) = 1 / x y , 1 / x \phi (x,y)=1/xy, 1/x , or 1 / y 1/y . For certain cases, they are also necessary conditions. Our results extend those of Boas and Móricz from the one-dimensional to the two-dimensional series.