### Abstract

It is shown that the maximal operator of the Riesz means with parameter α = (α1, …, αd) of a d-dimensional tempered distribution is bounded from the Hardy space Hp(Rd) to Lp(Rd) if max{d/(d 4- 1), l/(αk + 1), k = 1, …, d} < p < and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the d-dimensional Riesz means of a function f L1(Rd) converge a.e. to f. Moreover, we prove that the Riesz means are uniformly bounded on the spaces Hp(Rd) whenever max{d/(d+1), l/(αk + l), k = 1, …, d} < p <. Thus the Riesz means converge in Hp(Rd) norm to f E Hp(Rd). The same results are proved for the conjugate Riesz means and for d-dimensional Fourier series of distributions, too.

Original language | English |
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Pages (from-to) | 121-136 |

Number of pages | 16 |

Journal | Analysis (Germany) |

Volume | 20 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2000 |

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### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics

### Cite this

**Riesz means of d-dimensional fourier transforms and fourier series.** / Weisz, F.

Research output: Contribution to journal › Article

*Analysis (Germany)*, vol. 20, no. 2, pp. 121-136. https://doi.org/10.1524/anly.2000.20.2.121

}

TY - JOUR

T1 - Riesz means of d-dimensional fourier transforms and fourier series

AU - Weisz, F.

PY - 2000/1/1

Y1 - 2000/1/1

N2 - It is shown that the maximal operator of the Riesz means with parameter α = (α1, …, αd) of a d-dimensional tempered distribution is bounded from the Hardy space Hp(Rd) to Lp(Rd) if max{d/(d 4- 1), l/(αk + 1), k = 1, …, d} < p < and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the d-dimensional Riesz means of a function f L1(Rd) converge a.e. to f. Moreover, we prove that the Riesz means are uniformly bounded on the spaces Hp(Rd) whenever max{d/(d+1), l/(αk + l), k = 1, …, d} < p <. Thus the Riesz means converge in Hp(Rd) norm to f E Hp(Rd). The same results are proved for the conjugate Riesz means and for d-dimensional Fourier series of distributions, too.

AB - It is shown that the maximal operator of the Riesz means with parameter α = (α1, …, αd) of a d-dimensional tempered distribution is bounded from the Hardy space Hp(Rd) to Lp(Rd) if max{d/(d 4- 1), l/(αk + 1), k = 1, …, d} < p < and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the d-dimensional Riesz means of a function f L1(Rd) converge a.e. to f. Moreover, we prove that the Riesz means are uniformly bounded on the spaces Hp(Rd) whenever max{d/(d+1), l/(αk + l), k = 1, …, d} < p <. Thus the Riesz means converge in Hp(Rd) norm to f E Hp(Rd). The same results are proved for the conjugate Riesz means and for d-dimensional Fourier series of distributions, too.

UR - http://www.scopus.com/inward/record.url?scp=0039170945&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039170945&partnerID=8YFLogxK

U2 - 10.1524/anly.2000.20.2.121

DO - 10.1524/anly.2000.20.2.121

M3 - Article

AN - SCOPUS:0039170945

VL - 20

SP - 121

EP - 136

JO - Analysis (Germany)

JF - Analysis (Germany)

SN - 0174-4747

IS - 2

ER -