Abstract
We establish a criterion for the sum of a multiple (or one-dimensional) trigonometric series to belong to the Lebesgue space $$L_p([0,2\pi]^d)$$ for $$1<p<\infty$$ and $$d\ge 1$$ . We consider series whose coefficients satisfy a new general monotonicity condition, which does not reduce to any of the previously known forms of general monotonicity. In particular, we present examples of sequences that belong to this new class but are not contained in any of the previously studied classes of general monotone sequences. The results extend and refine the known criteria for the $$L_p$$ -integrability of the sums of multiple and one-dimensional trigonometric series.