Convergence and integrability of double trigonometric series with coefficients of bounded variation

Abstract:

We prove that if $c\left ( {j,k} \right ) \to 0$ as $\max \left ( {|j|,|k|} \right ) \to \infty$ and \[ \sum \limits _{j = - \infty }^\infty {\sum \limits _{k = - \infty }^\infty {\left | {{\Delta _{11}}c\left ( {j,k} \right )} \right | < \infty ,} } \] then the series $\sum \nolimits _{j = - \infty }^\infty {\sum \nolimits _{k = - \infty }^\infty {c\left ( {j,k} \right ){e^{i(jx + ky)}}} }$ converges both pointwise for every $\left ( {x,y} \right ) \in {\left ( {T\backslash \left \{ 0 \right \}} \right )^2}$ and in the ${L^p}\left ( {{T^2}} \right )$-metric for $0 < p < 1$, where $T$ is the one-dimensional torus. Both convergence statements remain valid for the three conjugate series under these same coefficient conditions.