中间四分之一康托测度的类傅里叶级数的几乎处处收敛性
Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure
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中文总结 AI 辅助
本文针对中间四分之一康托测度，证明其类傅里叶级数的极大算子为弱型(1,1)，进而得到该级数几乎处处收敛，解决了Strichartz的长期开放问题。
中文摘要 AI 辅助
1998年，Jorgensen和Pedersen构造了首个奇异连续谱测度的例子。他们证明，由迭代函数系{(x-1)/4, (x+1)/4}以等权重生成的自相似测度μ₁/₄是具有谱Λ₄的谱测度，Λ₄称为标准谱，定义为Λ₄ := {∑_{j=0}^{m-1}ε_j4^j : m≥1, ε_j∈{0,1}}。对f∈L¹(μ₁/₄)，设Sₙf是其关于Λ₄的第n个部分和。我们证明，对应的极大算子S*f = supₙ|Sₙf|是弱型(1,1)的，因此Sₙf在μ₁/₄的支撑集上μ₁/₄-几乎处处收敛于f，这解决了Strichartz提出的一个长期开放问题。
英文摘要
In 1998, Jorgensen and Pedersen constructed the first example of a singular continuous spectral measure. Precisely, they proved that the self-similar measure generated by the iterated function system $\{\frac{x-1}{4},\frac{x+1}{4}\}$ with equal weights, denoted by $μ_{1/4}$, is a spectral measure with a spectrum $Λ_4$, called the canonical spectrum,\[ Λ_4 := \set{ \sum_{j=0}^{m-1}\varepsilon_j4^j: m\ge 1,\ \varepsilon_j\in\{0,1\} }. \] For $f\in L^1(μ_{1/4})$, let $S_n f$ be the $n$-th partial sum of its Mock Fourier series with respect to $Λ_4$. We prove that the associated maximal operator $S^{\ast}f=\sup_n|S_n f|$ is of weak type $(1,1)$. Consequently, $S_n f\to f$ $μ_{1/4}$-almost everywhere on $\supp(μ_{1/4})$. This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.
发表机构
- School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）
- School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）
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