The martingale Hardy space Hp([0,1)2) and the classical Hardy space Hp(double-struck T sign2) 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 Hp to Lp (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.
- Atomic decomposition
- Hardy-Littlewood inequality
- Martingale and classical Hardy spaces
- Walsh functions
ASJC Scopus subject areas