Herz spaces and summability of Fourier transforms
Herz spaces and summability of Fourier transforms
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DOI：
10.1002/mana.200510604
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发表时间：
2008-03
影响因子：
1
通讯作者：
H. Feichtinger;F. Weisz
中科院分区：
文献类型：
--
作者：
H. Feichtinger;F. Weisz
考虑了Herz空间Kαp，r（？d）中函数的一般求和方法.证明了Hardy-Littlewood极大算子在某些临界情形下在Herz空间上的有界性。这意味着θ-平均σθ Tf的极大算子也在相应的Herz空间上有界，并且σθ Tf → f a. e.对所有的f ∈ K-d /p p，∞（？d）.此外，σθ Tf（x）在f ∈ K-d /p p，∞（φ d）的每个p-Lebesgue点处收敛于f（x）当且仅当θ的傅里叶变换在Herz空间Kd /p p ′，1（φ d）中.在Herz空间中也研究了θ-平均的范数收敛性。作为特殊情况，得到了加权Lp空间的一些结果.（© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA，魏因海姆）
A general summability method is considered for functions from Herz spaces Kαp,r (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 σθ T f is also bounded on the corresponding Herz spaces and σθ T f → f a.e. for all f ∈ K–d /p p,∞ (ℝd ). Moreover, σθ T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K–d /p p,∞ (ℝd ) if and only if the Fourier transform of θ is in the Herz space Kd /p p ′,1 (ℝd ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)