Carleson Measures for Slice Regular Hardy and Bergman Spaces in Quaternions

IF 1.1 2区 数学 Q2 MATHEMATICS, APPLIED
Wenwan Yang, Cheng Yuan
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引用次数: 0

Abstract

We study the quaternionic Carleson measure, which provides an embedding of the slice regular Hardy space \({\mathcal {H}}^p({\mathbb {B}})\) into \(L^s({\mathbb {B}}, \text {d}\mu )\) with \(s>p.\) A new criterion is needed for a finite positive Borel measure to be an \(({\mathcal {H}}^p({\mathbb {B}}),s)\)-Carleson measure, given by the uniform integrability of slice Cauchy kernels. It turns out that the symmetric box and the symmetric pseudo-hyperbolic disc are equivalent in the characterization of \(({\mathcal {H}}^p({\mathbb {B}}),s)\)-Carleson measures, while they are not when \(s=p.\) We further study the s-Carleson measure for slice regular Bergman spaces \({{\mathcal {A}}}^p({\mathbb {B}})\) for all indices sp. When \(s\ge p,\) our characterization relies on a close relation between the Carleson measure for Hardy and Bergman spaces and is primarily based on the slice Cauchy kernel, rather than the slice Bergman kernel. The advantage of the slice Cauchy kernel over the slice Bergman kernel is that, when restricted to any slice plane, the former, as a sum of two terms, transforms into a fractional linear transform, whereas the latter does not. This enables a locally uniform lower bound estimate for the slice Cauchy kernel, which is crucial in applications. In the case where \(s<p,\) we need to apply Khinchine’s inequality and a point-wise estimate for atoms in slice Bergman spaces based on the convex combination identity.

四元数中片正则Hardy和Bergman空间的Carleson测度
我们研究了四元数Carleson测度,它提供了将切片正则Hardy空间\({\mathcal {H}}^p({\mathbb {B}})\)用\(s>p.\)嵌入\(L^s({\mathbb {B}}, \text {d}\mu )\)的一种方法。利用切片柯西核的一致可积性给出了有限正Borel测度是\(({\mathcal {H}}^p({\mathbb {B}}),s)\) -Carleson测度的新判据。结果表明,对称盒和对称伪双曲盘在\(({\mathcal {H}}^p({\mathbb {B}}),s)\) -Carleson测度的表征中是等价的,而当\(s=p.\)时,它们不是等价的。我们进一步研究了片正则Bergman空间\({{\mathcal {A}}}^p({\mathbb {B}})\)的s-Carleson测度对于所有指标s, p。当\(s\ge p,\)时,我们的表征依赖于Hardy和Bergman空间的Carleson测度之间的密切关系,并且主要基于片Cauchy核。而不是切片伯格曼核。切片柯西核相对于切片伯格曼核的优点是,当限制在任何切片平面时,前者作为两项的和,变换成分数阶线性变换,而后者则不是。这使得切片柯西核的局部一致下界估计成为可能,这在应用中是至关重要的。在\(s<p,\)的情况下,我们需要应用Khinchine不等式和基于凸组合恒等式的切片Bergman空间中原子的点向估计。
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来源期刊
Advances in Applied Clifford Algebras
Advances in Applied Clifford Algebras 数学-物理:数学物理
CiteScore
2.20
自引率
13.30%
发文量
56
审稿时长
3 months
期刊介绍: Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.
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