### Abstract

Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from H_{P}(ℝ) to L_{P}(ℝ) (1/(α + 1) < p < ∞) and is of weak type (1,1), where H_{P}(ℝ) is the classical Hardy space. As a consequence we deduce that the Riesz means of a function f ∈ L_{1}(ℝ) converge a.e. to f. Moreover, we prove that the Riesz means are uniformly bounded on H_{P}(ℝ)) whenever 1/(α + 1) < p < ∞. Thus, in case f ∈ H_{p}(ℝ), the Riesz means converge to f in H_{p}(ℝ) norm (1/(α + 1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.

Original language | English |
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Pages (from-to) | 253-270 |

Number of pages | 18 |

Journal | Studia Mathematica |

Volume | 131 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 1998 |

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### Keywords

- Atomic decomposition
- Fourier transforms
- Hardy spaces
- Interpolation
- P-atom
- Riesz means

### ASJC Scopus subject areas

- Mathematics(all)