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A holomorphic extension result. (English) Zbl 0865.42008

The paper establishes a Fourier series theory on star-shaped Lipschitz curves and proves that if \((b_n)\) is the restriction to positive integers of a function \(b\) holomorphically defined in a sector \(\{z\in \mathbb{C}: |\text{arg} (z)|< \omega\}\), \(\omega\in (0, {\pi \over 2})\), satisfying \(|b(z) |< C|z|^s\) at infinity and bounded at the origin, then the power series \(\varphi (z)= \sum^\infty_{n=1} b_n z^n\), \(|z|< 1\), is holomorphically extensible to a heart-shaped region containing the set \(\{z\in \mathbb{C}: z\neq 1\), \(|z|= 1\}\) in its interior and dominated by \({C \over {|1-z |^{1+s}}}\) for \(z\) near 1 in the region, where \(-\infty <s< \infty\). The converse result also holds. The results can be extended to Laurent series as well. The theory has applications to singular and fractional integrals on closed Lipschitz curves.
Reviewer: T.Qian (Armidale)

MSC:

42B15 Multipliers for harmonic analysis in several variables
30B40 Analytic continuation of functions of one complex variable
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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