Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces

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• 作者：Ferenc Weisz (1)
  
• 关键词：Wiener amalgam spaces ; Herz spaces ; Strong Hardy ; Littlewood maximal function ; $$\theta $$ θ ; summability ; Lebesgue points. ; Primary 42B08 ; 46E30 ; Secondary 42B30 ; 42A38
• 刊名：Monatshefte f篓鹿r Mathematik
• 出版年：2014
• 出版时间：September 2014
• 年：2014
• 卷：175
• 期：1
• 页码：143-160
• 全文大小：201 KB
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  13. Weisz, F.: Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2002) CrossRef
  14. Weisz, F.: Summability of Multi-Dimensional Trigonometric Fourier Series, Surv. Appr Theory, 7, 1-79 (2012)
  15. Zygmund, A.: Trigonometric Series, 3rd edn. Cambridge Press, London (2002)
  
• 作者单位：Ferenc Weisz (1)
  
  1. Department of Numerical Analysis, E?tv?s L. University, Pázmány P. sétány 1/C., Budapest?, 1117, Hungary
  
• ISSN：1436-5081

文摘

New multi-dimensional Wiener amalgam spaces \(W_c(L_p,\ell _\infty )(\mathbb{R }^d)\) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the \(\theta \) -summability of multi-dimensional Fourier transforms is studied. It is proved that if the Fourier transform of \(\theta \) is in a suitable Herz space, then the \(\theta \) -means \(\sigma _T^\theta f\) converge to \(f\) a.e. for all \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) . Note that \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset W_c(L_r,\ell _\infty )(\mathbb{R }^d) \supset L_r(\mathbb{R }^d)\) and \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset L_1(\log L)^{d-1}(\mathbb{R }^d)\) , where \(1 . Moreover, \(\sigma _T^\theta f(x)\) converges to \(f(x)\) at each Lebesgue point of \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) .