Boundary values and Poisson integral representations

1984 ◽  
pp. 79-96
Author(s):  
Henrik Schlichtkrull
Keyword(s):  
Integral Representations ◽  
Poisson Integral ◽  
Boundary Values

Corrigenda: Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values

1984 ◽  
Vol 96 (1-2) ◽  
pp. 179-179 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Compact Support ◽  
Integral Representations ◽  
Poisson Integral ◽  
Boundary Values

Page 56: In the estimate , C1 = CSn/(2πσ)n, h = (2πσ)−1 where σ depends on y. Therefore, {fy(t)} may not be bounded in as y → 0, y ∈ C′.

Integral Representations and Continuous Projectors in Some Spaces of Analytic and Pluriharmonic Functions

2008 ◽  
Vol 15 (4) ◽  
pp. 739-752
Author(s):  
Gigla Oniani ◽  
Lamara Tsibadze
Keyword(s):  
Integral Representations ◽  
Fractional Integrals ◽  
Boundary Values ◽  
Pluriharmonic Functions

Abstract We consider analytic and pluriharmonic functions belonging to the classes 𝐵𝑝(Ω) and 𝑏𝑝(Ω) and defined in the ball . The theorems established in the paper make it possible to obtain some integral representations of functions of the above-mentioned classes. The existence of bounded projectors from the space 𝐿(ρ, Ω) into the space 𝐵𝑝(Ω) and from the space 𝐿(ρ, Ω) into the space 𝑏𝑝(Ω) is proved. Also, consideration is given to the existence of boundary values of fractional integrals of functions of the spaces 𝐵𝑝(Ω) and 𝑏𝑝(Ω).

Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values

1981 ◽  
Vol 91 (1-2) ◽  
pp. 49-62 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Analytic Function ◽  
Growth Condition ◽  
Compact Support ◽  
Analytic Functions ◽  
Convex Cone ◽  
Integral Representations ◽  
Boundary Values ◽  
Cauchy Integral ◽  
Boundedness Condition ◽  
Poisson Integrals

SynopsisUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.

scholarly journals A new transform method II: the global relation and boundary-value problems in polar coordinates

2010 ◽  
Vol 466 (2120) ◽  
pp. 2283-2307 ◽  
Author(s):  
E. A. Spence ◽  
A. S. Fokas
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Integral Representations ◽  
New Method ◽  
Polar Coordinates ◽  
Nonlinear Pdes ◽  
Boundary Values ◽  
Transform Method ◽  
Global Relation ◽  
The Fourier Transform

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In a previous paper, a simplified way of obtaining these representations was presented. In the current paper, first, the second ingredient of the new method, namely the derivation of the so-called ‘global relation’ (GR)—an equation involving transforms of the boundary values—is presented. Then, using the GR as well as the IR derived in the previous paper, certain BVPs in polar coordinates are solved. These BVPs elucidate the fact that this method has substantial advantages over the classical transform method.

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