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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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