Slice Lebesgue Measure of Quaternions

Ren, Guangbin; Xu, Zhenghua

doi:10.1007/s00006-015-0578-1

Cited by 6 publications

(3 citation statements)

References 16 publications

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“…This convex combination identity is also quite useful for other purposes. For instance, it provides an effective approach to a quaternionic version of a well-known Forelli-Rudin estimate, which will play a fundamental role in the theory of various spaces of slice regular functions [30].…”

Section: Growth Distortion and Covering Theorems For Slice Regular Fmentioning

confidence: 99%

Growth and distortion theorems for slice monogenic functions

Ren

Wang

2017

Pacific J. Math.

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Abstract. The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc D ⊂ C in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.

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Section: Growth Distortion and Covering Theorems For Slice Regular Fmentioning

confidence: 99%

Growth and distortion theorems for slice monogenic functions

Ren

Wang

2017

Pacific J. Math.

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“…In [10], Krantz extended this result to harmonic functions. See also the version of the Hardy-Littlewood theorem for quaternionic slice regular functions [17]. As an application of Theorem 1, we first establish the Hardy-Littlewood theorem in the infinite-dimensional Hilbert space.…”

Section: Applicationsmentioning

confidence: 99%

Bloch type spaces on the unit ball of a Hilbert space

2018

Czech.Math.J.

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“…同时, 我们对单位圆盘 D 上的正规化双全纯函数的 Slice 正则延拓建立了相应的增长定理与掩盖定理 [32] . 我们还研究了正则函数理论中的函数空间理论, 建立了相应的 Forelli-Rudin 估计 [33] .…”

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Recent development of the theory of Slice regular functions

Ren¹,

Wang²,

Xu³

2015

Sci. Sin.-Math.

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Theory of Slice regular functions is a natural generalization of complex analysis from complex numbers to quaternions. It was introduced initially by Gentili and Struppa in 2006, and has been extensively studied and has found its elegant applications to functional calculus for quaternionic linear operators, operator theory, Schur analysis and quaternionic quantum mechanics. Meanwhile, the notion of Slice regularity was also extended to functions of an octonionic variable and to the setting of Clifford algebras as well as to the setting of alternative real algebras. In this survey, we shall focus mainly on our recent results in the theory of Slice regular functions. Firstly, we establish the Borel-Carathéodory inequality and the sharp growth and distortion theorems for Slice regular extensions of normalized biholomorphic functions on the unit disc in the setting of quaternions. In addition, we study the boundary behavior of Slice regular functions and obtain the Julia lemma, the Julia-Carathéodory theorem as well as the boundary Schwarz lemma. In particular, we find that the boundary Schwarz lemma does not ensure the Slice derivative of a Slice regular self-mappings of the open unit ball B ⊂ H at its boundary fixed point to be necessarily a real number, in contrast to that in the complex case. Finally, we study theory of function spaces of Slice regular functions and establish the Forelli-Rudin type estimates.

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Slice Lebesgue Measure of Quaternions

Cited by 6 publications

References 16 publications

Growth and distortion theorems for slice monogenic functions

Growth and distortion theorems for slice monogenic functions

Bloch type spaces on the unit ball of a Hilbert space

Recent development of the theory of Slice regular functions

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