### Abstract

The d-dimensional Hardy spaces H_{p}(T^{d1} × . . . × T^{dk}) (d = d_{1} + . . . + d_{k}) and a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θ_{j} having integrable Fourier transforms. Under some conditions on θ_{j} we show that the maximal operator of the θ-means of a distribution is bounded from H_{p} (T^{d1} × . . . × T^{dk}) to L_{p} (T^{d}) where p_{0} < p < ∞ and p^{0} < 1 is depending only on the functions θ_{j}. By an interpolation theorem we get that the maximal operator is also of weak type (H^{#i}_{1}, L_{1}) (i = 1, . . . , k) where the Hardy space H^{#i}_{1} is defined by a hybrid maximal function and H^{#i}_{1} = L_{1} if k = 1. As a consequence we obtain that the θ-means of a function f ∈ H^{#i}_{1} ⊃ L(log L)^{k-1} converge a. e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L_{1}. Moreover, we prove that the θ-means are uniformly bounded on the spaces H_{p} (T^{d1} × . . . × T^{dk}) whenever p_{0} < p < ∞, thus the θ-means converge to f in H_{p} (T^{d1} × . . . × T^{dk}) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.

Original language | English |
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Pages (from-to) | 159-180 |

Number of pages | 22 |

Journal | Mathematische Nachrichten |

Volume | 230 |

DOIs | |

Publication status | Published - Jan 1 2001 |

### Keywords

- Atomic decomposition
- Fourier transforms
- Hardy spaces
- Interpolation
- p-atom
- θ-summation

### ASJC Scopus subject areas

- Mathematics(all)