Geodesic
Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients
Studia Mathematica, Tome 125 (1997) no. 3, pp. 231-246

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Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $(∑_{k=1}^∞ ∑_{j=1}^∞ |f̂(k,j)|^{p}(kj)^{p-2})^{1/p} ≤ C_p∥f∥_{H^p_{**}}$ (1/2 p≤2) where f belongs to the Hardy space $H_{**}^p (G_m × G_s)$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.
DOI : 10.4064/sm-125-3-231-246
Keywords: two-parameter martingales and Hardy spaces, rectangle p-atoms, Vilenkin functions, Hardy-Littlewood inequality

Péter Simon 1 ;  1

1 
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Péter Simon;  . Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients. Studia Mathematica, Tome 125 (1997) no. 3, pp. 231-246. doi: 10.4064/sm-125-3-231-246

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