The Boundary Behavior of Holomorphic Functions: Global and Local Results

Krantz, Steven G.

doi:10.4310/ajm.2007.v11.n2.a2

Cited by 6 publications

(3 citation statements)

References 31 publications

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“…In addition to the problems mentioned above, some other boundary value problems related to Theorem 1 can be found in the work of Bochner [Boc44], Weinstock [Wei69], Stein [Ste70,Ste73], Jacewicz [Jac73], and Krantz [Kra80,Kra07]. These works prove a number results about the behavior of holomorphic functions on domains with smooth boundaries in C n , but the point of view taken in [Boc44,Wei69,Ste70,Ste73,Jac73,Kra80,Kra07] is different than the one taken in this work. They start with a holomorphic function G defined on a domain D and make conclusions about the G near or on the boundary ∂D.…”

Section: Introductionmentioning

confidence: 78%

Holomorphic extension on product Lipschitz surfaces in two complex variables

Hart¹,

Monguzzi²

2014

Preprint

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In this work we prove a new L p holomorphic extension result for functions defined on product Lipschitz surfaces with small Lipschitz constants in two complex variables. We define biparameter and partial Cauchy integral operators that play the role of boundary values for holomorphic functions on product Lipschitz domain. In the spirit of the application of David-Journé-Semmes and Christ's T b theorem to the Cauchy integral operator, we prove a biparameter T b theorem and apply it to prove L p space bounds for the biparameter Cauchy integral operator. We also prove some new biparameter Littlewood-Paley-Stein estimates and use them to prove the biparameter T b theorem.

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Section: Introductionmentioning

confidence: 78%

Holomorphic extension on product Lipschitz surfaces in two complex variables

Hart¹,

Monguzzi²

2014

Preprint

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“…(For a thoughtful treatment on these results, see [53].) Since then, criteria on local existence of non-tangential boundary limits of harmonic functions in many different contexts, in terms of non-tangential boundedness or one-side non-tangential boundedness or finiteness of non-tangential area integral have been intensively studied (see, among others, [2]- [5], [8], [18], [20]- [22], [24]- [28], [30], [36]- [38], [40]- [42], [44], [58]). Various aspects associated with the Fatou theory in one and several complex variables are contained in the most recent paper [10].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local boundary behaviour and the area integral of generalized harmonic functions associated with root systems

Jiu¹,

Li²

2022

Preprint

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The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space R d associated with a root system. The aim of the paper is to study local boundary behaviour of generalized harmonic functions associated with the Dunkl operators. We introduce a Lusin-type area integral operator S by means of Dunkl's generalized translation and the Dunkl operators. The main results are on characterizations of local existence of non-tangential boundary limits of a generalized harmonic function u in the upper half-space R d+1 + associated with the Dunkl operators, and for a subset E of R d invariant under the reflection group generated by the root system, the equivalence of the following three assertions are proved: (i) u has a finite non-tangential limit at (x, 0) for a.e. x ∈ E; (ii) u is non-tangentially bounded for a.e. x ∈ E; (iii) (Su)(x) is finite for a.e. x ∈ E.

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“…A generalization of the theorem of Spencer [42] to several variables was obtained in Stein [43]. Since then, criteria on existence of non-tangential boundary limits of harmonic functions in many different contexts, in terms of non-tangential boundedness or one-side non-tangential boundedness or finiteness of area integrals have been intensively studied; see, for example, [1][2][3][4]7,[14][15][16][17][18][19][20][21][22]24,[32][33][34][35][36][37]39] and [46].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

sci¹

2020

ATA

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We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R 2 + = R × (0, ∞), where (D f)(x) = f (x) + (λ/x)[ f (x) − f (−x)] for given λ ≥ 0. A C 2 function u in R 2 + is said to be λ-harmonic if (D 2 x + ∂ 2 y)u = 0. For a λ-harmonic function u in R 2 + and for a subset E of ∂R 2 + = R symmetric about y-axis, we prove that the following three assertions are equivalent: (i) u has a finite non-tangential limit at (x, 0) for a.e. x ∈ E; (ii) u is non-tangentially bounded for a.e. x ∈ E; (iii) (Su)(x) < ∞ for a.e. x ∈ E, where S is a Lusin-type area integral associated with the Dunkl operator D.

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The Boundary Behavior of Holomorphic Functions: Global and Local Results

Cited by 6 publications

References 31 publications

Holomorphic extension on product Lipschitz surfaces in two complex variables

Holomorphic extension on product Lipschitz surfaces in two complex variables

Local boundary behaviour and the area integral of generalized harmonic functions associated with root systems

Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

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