A NEW BIHARMONIC KERNEL FOR THE UPPER HALF PLANE

Title & Authors
A NEW BIHARMONIC KERNEL FOR THE UPPER HALF PLANE
Abkar, Ali;

Abstract
We introduce a new biharmonic kernel for the upper half plane, and then study the properties of its relevant potentials, such as the convergence in the mean and the boundary behavior. Among other things, we shall see that Fatou's theorem is valid for these potentials, so that the biharmonic Poisson kernel resembles the usual Poisson kernel for the upper half plane.
Keywords
Poisson kernel;biharmonic function;biharmonic Green function;convergence in the mean;Fatou's theorem;
Language
English
Cited by
1.
Biharmonic Green Functions on Homogeneous Trees, Mediterranean Journal of Mathematics, 2009, 6, 3, 249
References
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A. Abkar and H. Hedenmalm, A Riesz representation formula for super-bi- harmonic functions, Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 305-324

2.
P. R. Garabedian, Partial Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964

3.
J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981

4.
H. Hedenmalm, A computation of Green functions for the weighted biharmonic operators ${\Delta}|z|^{-2{\alpha}}{\Delta}$ with ${\alpha}>-1$ Duke Math. J. 75 (1994), no. 1, 51-78