Higher-Dimensional Boundary Analogs of the Morera Theorem in Problems of Analytic Continuation of Functions

Kytmanov, Alexander M.; Myslivets, S. G.

doi:10.1023/b:joth.0000020707.81285.06

Cited by 22 publications

(15 citation statements)

References 26 publications

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“…Stout [19] observed that complex lines intersecting a given nonlinear algebraic curve are sufficient for C 1 -functions. Other results in this direction and extended references can be found in [11], [13], [14], [15].…”

Section: Introduction and The Main Resultsmentioning

confidence: 68%

Analog of a theorem of forelli for boundary values of holomorphic functions on the unit ball of ℂ n

Agranovsky

2011

JAMA

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Section: Introduction and The Main Resultsmentioning

confidence: 68%

Analog of a theorem of forelli for boundary values of holomorphic functions on the unit ball of ℂ n

Agranovsky

2011

JAMA

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“…Lemma 4. If a function f ∈ C(∂D) has a one-dimensional holomorphic continuation property along almost all complex lines l ∈ L Γ and the connected components of intersection D ∩ l are domains with a strengthened Nevanlinna property, then the equalities ( 9) are true for all multiindices α with α > 0 and for all j = 1, . .…”

Section: Theorem 5 ([11]mentioning

confidence: 99%

On the Families of Complex Lines Sufficient for Holomorphic Continuation of Functions Defined on a Domain Boundary

Kytmanov¹,

Myslivets²

2013

Complex Analysis and Dynamical Systems V

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This paper presents selected results on the study of the onedimensional property of holomorphic continuations of functions defined on the boundary of a bounded domain in C n. This paper presents selected results on holomorphic continuation of functions defined on the boundary of a bounded domain D ⊂ C n , n > 1, into this domain. We consider functions with a one-dimensional holomorphic continuation property along families of complex lines. On a complex plane C, results for functions with a one-dimensional holomorphic continuation property are trivial; therefore, results of this work are essentially multidimensional. The first result concerning the subject of our interest was obtained by M. L. Agranovsky and R. E. Val'ski in [2], who studied functions with a onedimensional holomorphic continuation property in a sphere. Their approach was based on the automorphism group properties of a sphere. E.L. Stout in [12] used complex Radon transformation to generalize the Agranovsky and Val'ski theorem for an arbitrary bounded domain with a smooth boundary. An alternative proof of the Stout theorem was obtained by A.M. Kytmanov in [3] by applying the Bochner-Martinelli integral. The idea of using integral representations (Bochner-Martinelli, Cauchy-Fantappiè, logarithmic residue) has turned out to be useful for studying of functions with one-dimensional holomorphic continuation property along complex lines and curves [7, 8]. A review of the results on the subject under discussion can be found in [9]. Let D be a bounded domain in C n , n > 1, with connected smooth boundary ∂D (of a class C 2). Let us formulate E.L. Stout's result [12].

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“…Various other families are given in [5,9,11,14,15]. In [6,10,16,17] it is shown that for holomorphic extension of continuous functions defined on the boundary of the ball + 1 points inside the ball not lying on a complex hyperplane is enough.…”

Section: Introductionmentioning

confidence: 99%

On functions with one‐dimensional holomorphic extension property in circular domains

Kytmanov

Myslivets

2019

Mathematische Nachrichten

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Higher-Dimensional Boundary Analogs of the Morera Theorem in Problems of Analytic Continuation of Functions

Cited by 22 publications

References 26 publications

Analog of a theorem of forelli for boundary values of holomorphic functions on the unit ball of ℂ n

Analog of a theorem of forelli for boundary values of holomorphic functions on the unit ball of ℂ n

On the Families of Complex Lines Sufficient for Holomorphic Continuation of Functions Defined on a Domain Boundary

On functions with one‐dimensional holomorphic extension property in circular domains

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