### Abstract

A general summability method is considered for functions from Herz spaces K_{p,r}^{α} (ℝ^{d}). The boundedness of the Hardy-Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ-means σ_{T}^{θ} f is also bounded on the corresponding Herz spaces and σ_{T}^{θ} f → f a.e. for all f ∈ K_{p,∞}^{-d/p} (ℝ^{d}). Moreover, σ_{T}^{θ}f(x) converges to f(x) at each p-Lebesgue point of f ∈ K_{p,∞}^{-d/p} (ℝ^{d}) if and only if the Fourier transform of θ is in the Herz space K _{p′,1}^{d/p} (ℝ^{d}). Norm convergence of the θ-means is also investigated in Herz spaces. As special cases some results are obtained for weighted L_{p} spaces.

Original language | English |
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Pages (from-to) | 309-324 |

Number of pages | 16 |

Journal | Mathematische Nachrichten |

Volume | 281 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2008 |

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### Keywords

- θ-summability of Fourier series
- Hardy-Littlewood maximal function
- Herz spaces
- Lebesgue points
- Weighted L spaces
- Weighted Wiener amalgam spaces

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Nachrichten*,

*281*(3), 309-324. https://doi.org/10.1002/mana.200510604

**Herz spaces and summability of Fourier transforms.** / Feichtinger, Hans G.; Weisz, F.

Research output: Contribution to journal › Article

*Mathematische Nachrichten*, vol. 281, no. 3, pp. 309-324. https://doi.org/10.1002/mana.200510604

}

TY - JOUR

T1 - Herz spaces and summability of Fourier transforms

AU - Feichtinger, Hans G.

AU - Weisz, F.

PY - 2008/3

Y1 - 2008/3

N2 - A general summability method is considered for functions from Herz spaces Kp,rα (ℝd). The boundedness of the Hardy-Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ-means σTθ f is also bounded on the corresponding Herz spaces and σTθ f → f a.e. for all f ∈ Kp,∞-d/p (ℝd). Moreover, σTθf(x) converges to f(x) at each p-Lebesgue point of f ∈ Kp,∞-d/p (ℝd) if and only if the Fourier transform of θ is in the Herz space K p′,1d/p (ℝd). Norm convergence of the θ-means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces.

AB - A general summability method is considered for functions from Herz spaces Kp,rα (ℝd). The boundedness of the Hardy-Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ-means σTθ f is also bounded on the corresponding Herz spaces and σTθ f → f a.e. for all f ∈ Kp,∞-d/p (ℝd). Moreover, σTθf(x) converges to f(x) at each p-Lebesgue point of f ∈ Kp,∞-d/p (ℝd) if and only if the Fourier transform of θ is in the Herz space K p′,1d/p (ℝd). Norm convergence of the θ-means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces.

KW - θ-summability of Fourier series

KW - Hardy-Littlewood maximal function

KW - Herz spaces

KW - Lebesgue points

KW - Weighted L spaces

KW - Weighted Wiener amalgam spaces

UR - http://www.scopus.com/inward/record.url?scp=55449098127&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55449098127&partnerID=8YFLogxK

U2 - 10.1002/mana.200510604

DO - 10.1002/mana.200510604

M3 - Article

AN - SCOPUS:55449098127

VL - 281

SP - 309

EP - 324

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

IS - 3

ER -