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On the coefficients of multiple Fourier series in $$L_p$$-spaces. (English. Russian original) Zbl 0976.42018
Izv. Math. 64, No. 1, 93-120 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 1, 95-122 (2000).
The author studies new function spaces and uses interpolation methods to obtain the analogues of the classical Hardy-Littlewood theorem and the Hardy-Littlewood-Paley inequalities for multiple orthogonal series, in particular, multiple trigonometric series. For example, he considers the space $$L^*_{pq}(\mathbb{R}^n)$$ endowed with the norm $\|f\|:= \Biggl(\int^\infty_0\cdots \Biggl(\int^\infty_0 t^{1/p_1}_1\cdots t^{1/p_n}_n|f(t_1,\dots, t_n)|^{q_1} {dt_1\over t_1}\Biggr)^{q_2/q_1}\cdots {dt_n\over t_n}\Biggr)^{1/q_n}, 0< p_j,q_j\leq\infty,$ and in case $$q=\infty$$, $$(\int^\infty_0(\cdot)^q{dt\over t})^{1/q}$$ means $$\sup_{t>0}(\cdot)$$; nonincreasing functions $$f(t_1,\dots, t_n)$$ in the extended sense on the cube $$[0,1]^n$$, which are defined by the requirement that $|f(x)|\leq c|x_1|^{-1}\cdots|x_n|^{-1}\Biggl|\int_{[0,x]} f(t) dt\Biggr|$ for any $$x= (x_1,\dots, x_n)\in [0,1]^n$$, where $$c$$ is a constant, etc.

##### MSC:
 42C15 General harmonic expansions, frames 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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