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Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

100%

Simon P., Weisz F.

Studia Mathematica

1997

tom 125

nr 3

231-246

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.

Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series

88%

Weisz F.

Studia Mathematica

1995-1996

tom 117

nr 2

173-194

The martingale Hardy space $H_p([0,1)^2)$ and the classical Hardy space $H_p(𝕋^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a Marcinkiewicz-Zygmund type inequality is obtained for BMO spaces.

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2 Studia Mathematica

2 Weisz F.

1 Simon P.

1 1997

1 1996

Wyniki wyszukiwania

Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series