On the coefficients of multiple Fourier series in $L_p$-spaces

In this paper we use new function spaces and interpolation methods to study the dependence of the properties of summable multiple Fourier series on their coefficients. We obtain theorems for multiple orthogonal series that reinforce the Hardy–Littlewood theorem for trigonometric series. We prove inequalities of Hardy–Littlewood–Paley type for multiple orthogonal series that refine certain well-known inequalities of this kind.