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Inequalities relative to two-parameter Vilenkin-Fourier coefficients. (English) Zbl 0728.60046
Summary: The inequality \[ (*)\quad (\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}(nm)^{p-2}| \hat f(n,m)|^ p)^{1/p}\leq C_ p\| f\|_{H^-_ p}\quad (0<p\leq 2) \] and its dual inequality are proved for two-parameter Vilenkin-Fourier coefficients and for two-parameter martingale Hardy spaces \(H^-_ p\) defined by means of the \(L^ p\)-norm of the conditional quadratic variation. The inequality (*) is extended to bounded Vilenkin systems and monotone coefficients for all p. The converse of the last inequality is also true for all p. From this it follows easily that under the same conditions the two-parameter Vilenkin- Fourier series of an arbitrary \(L^ p\) function \((p>1)\) converges a.e. to that function.

MSC:
60G42 Martingales with discrete parameter
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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