Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2020-11-18 , DOI: 10.1007/s00041-020-09798-8 Reuben Wheeler
For a general compact variety \(\Gamma \) of arbitrary codimension, one can consider the \(L^p\) mapping properties of the Bochner–Riesz multiplier
$$\begin{aligned} m_{\Gamma , \alpha }(\zeta ) \ = \ \mathrm{dist}(\zeta , \Gamma )^{\alpha } \phi (\zeta ) \end{aligned}$$where \(\alpha > 0\) and \(\phi \) is an appropriate smooth cutoff function. Even for the sphere \(\Gamma = {{\mathbb {S}}}^{N-1}\), the exact \(L^p\) boundedness range remains a central open problem in Euclidean harmonic analysis. In this paper we consider the \(L^p\) integrability of the Bochner–Riesz convolution kernel for a particular class of varieties (of any codimension). For a subclass of these varieties the range of \(L^p\) integrability of the kernels differs substantially from the \(L^p\) boundedness range of the corresponding Bochner–Riesz multiplier operator.
中文翻译：
关于一类Bochner–Riesz核的$$ L ^ p $$ L p可积性的一个注记
对于任意余维的一般紧致变体\（\ Gamma \），可以考虑Bochner–Riesz乘子的\（L ^ p \）映射属性
$$ \ begin {aligned} m _ {\ Gamma，\ alpha}（\ zeta）\ = \ \ mathrm {dist}（\ zeta，\ Gamma）^ {\ alpha} \ phi（\ zeta）\ end {aligned} $$其中\（\ alpha> 0 \）和\（\ phi \）是合适的平滑截止函数。即使对于球体（（Gamma = {{\ mathbb {S}}} ^ {N-1} \），精确的\（L ^ p \）有界范围仍然是欧几里德谐波分析中的一个中心开放问题。在本文中，我们考虑了Bochner-Riesz卷积核的（（L ^ p \））对于特定类别的变种（任意维）的\（L ^ p \）可积性。对于这些变体的一个子类，内核的\（L ^ p \）可积范围与相应的Bochner–Riesz乘子算子的\（L ^ p \）有界范围实质上不同。
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