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harmonic morphisms; Kelvin transform; Weinstein operator
The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator $W_k:=\Delta+\frac k{x_n}\frac{\partial}{\partial x_n}$ on $\Bbb R^n$ is proved. In this note there is shown that in the cases $k\neq 0$, $k\neq 2$ no other transforms of this kind exist and for case $k=2$, all such transforms are described.
[1] Kellogg O.D.: Foundation of Potential Theory. Springer-Verlag, Berlin, 1929 (reissued 1967). MR 0222317
[2] Leutwiler H.: On the Appell transformation. in: Potential Theory (ed. J. Král et al.), Plenum Press, New York, 1987, pp.215-222. MR 0986298 | Zbl 0685.35006
[3] Brzezina M.: Appell type transformation for the Kolmogorov type operator. Math. Nachr. 169 (1994), 59-67. MR 1292797
[4] Brzezina M., Šimůnková M.: On the harmonic morphism for the Kolmogorov type operators. in: Potential Theory - ICPT 94, Walter de Gruyter, Berlin, 1996, pp.341-357. MR 1404718
[5] Akin Ö., Leutwiler H.: On the invariance of the solutions of the Weinstein equation under Möbius transformations. in: Classical and Modern Potential Theory and Applications (ed. K. GowriSankaran et al.), Kluwer Academic Publishers, 1994, pp.19-29. MR 1321603 | Zbl 0869.31005
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