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  • Meromorphic extensions from small families of circles and holomorphic extensions from spheres
    Globevnik, Josip, 1945-
    Let ▫$\mathbb{B}$▫ be the open unit ball in ▫$\mathbb{C}^2$▫ and let ▫$a, b, c$▫ be three points in ▫$\mathbb{C}^2$▫ which do not lie in a complex line, such that the complex line through ▫$a, b$▫ ... meets ▫$\mathbb{B}$▫ and such that if one of the points ▫$a, b$▫ is in ▫$\mathbb{B}$▫ and the other in ▫$\mathbb{C}^2 \setminus \overline {\mathbb{B}}$▫ then ▫$\langle a\vert b \rangle \not = 1$▫ and such that at least one of the numbers ▫$\langle a\vert c\rangle$▫, ▫$\langle b \vert c \rangle$▫ is different from ▫$1$▫. We prove that if a continuous function ▫$f$▫ on ▫$b\mathbb{B}$▫ extends holomorphically into ▫$\mathbb{B}$▫ along each complex line which meets ▫$\{ a, b, c\}$▫, then ▫$f$▫ extends holomorphically through ▫$\mathbb{B}$▫. This generalizes the recent result of L. Baracco who proved such a result in the case when the points ▫$a, b, c$▫ are contained in ▫$\mathbb{B}$▫. The proof is quite different from the one of Baracco and uses the following one-variable result, which we also prove in the paper: Let ▫$\Delta $▫ be the open unit disc in ▫$\mathbb{C}$▫. Given ▫$\alpha \in \Delta$▫ let ▫$\mathcal {C}_\alpha$▫ be the family of all circles in ▫$\Delta$▫ obtained as the images of circles centered at the origin under an automorphism of ▫$\Delta$▫ that maps ▫$0$▫ to ▫$\alpha$▫. Given ▫$\alpha , \beta \in \Delta$▫, ▫$\alpha \not = \beta$▫, and ▫$n \in \mathbb{N}$▫, a continuous function ▫$f$▫ on ▫$\overline {\Delta }$▫ extends meromorphically from every circle ▫$\Gamma \in \mathcal{C}_\alpha \cup \mathcal{C}_\beta$▫ through the disc bounded by ▫$\Gamma$▫ with the only pole at the center of ▫$\Gamma$▫ of degree not exceeding ▫$n$▫ if and only if ▫$f$▫ is of the form ▫$f(z) = a_0(z) + a_1(z){\overline z} +\cdots + a_n(z){\overline z}^n$▫ ▫$(z \in \Delta)$▫ where the functions ▫$a_j, \; 0 \leq j \leq n$▫, are holomorphic on ▫$\Delta $▫.
    Source: Transactions of the American Mathematical Society. - ISSN 0002-9947 (Vol. 364, no. 11, 2012, str. 5857-5880)
    Type of material - article, component part
    Publish date - 2012
    Language - english
    COBISS.SI-ID - 16364633

    Link(s):

    http://dx.doi.org/10.1090/S0002-9947-2012-05669-8

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