A series expansion for generalized harmonic functions

Analysis and Mathematical Physics - Tập 11 - Trang 1-28 - 2021
Markus Klintborg1, Anders Olofsson1
1Mathematics, Faculty of Science, Centre for Mathematical Sciences, Lund University, Lund, Sweden

Tóm tắt

We consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.

Tài liệu tham khảo

Ahern, P., Bruna, J., Cascante, C.: \(H^p\)-theory for generalized \({\cal{M}}\)-harmonic functions in the unit ball. Indiana Univ. Math. J. 45, 103–135 (1996) Ahern, P., Cascante, C.: Exceptional sets for Poisson integrals of potentials on the unit sphere in \({\mathbb{C}^n}\), \(p\le 1\). Pac. J. Math. 153, 1–13 (1992) Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999) Astala, K., Päivärinta, L.: Calderón’s inverse conductivity problem in the plane. Ann. Math. 163, 265–299 (2006) Behm, G.: Solving Poisson’s equation for the standard weighted Laplacian in the unit disc (2014). arXiv:1306.2199v2 [math.AP] Borichev, A., Hedenmalm, H.: Weighted integrability of polyharmonic functions. Adv. Math. 264, 464–505 (2014) Calderón, A.P.: On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp. 65–73, Soc. Brasil. Mat., Rio de Janeiro, : reprinted in Comput. Appl. Math. 25(2006), 133–138 (1980) Carlsson, M., Wittsten, J.: The Dirichlet problem for standard weighted Laplacians in the upper half plane. J. Math. Anal. Appl. 436, 868–889 (2016) Chen, S., Vuorinen, M.: Some properties of a class of elliptic partial differential operators. J. Math. Anal. Appl. 431, 1124–1137 (2015) Chen, X., Kalaj, D.: A representation theorem for standard weighted harmonic mappings with an integer exponent and its applications. J. Math. Anal. Appl. 444, 1233–1241 (2016) Conway, J.B.: Functions of One Complex Variable, second edition, Graduate Texts in Mathematics, vol. 11. Springer, New York (1978) Garabedian, P.R.: A partial differential equation arising in conformal mapping. Pac. J. Math. 1, 485–524 (1951) Geller, D.: Some results in \(H^p\) theory for the Heisenberg group. Duke Math. J. 47, 365–390 (1980) Hedenmalm, H.: On the uniqueness theorem of Holmgren. Math. Z. 281, 357–378 (2015) Katznelson, Y.: An Introduction to Harmonic Analysis. Dover, New York City (1976) Li, P., Wang, X.: Lipschitz continuity of \(\alpha \)-harmonic functions. Hokkaido Math. J. 48, 85–97 (2019) Li, P., Wang, X., Xiao, Q.: Several properties of \(\alpha \)-harmonic functions in the unit disk. Monatsh. Math. 184, 627–640 (2017) Olofsson, A.: Differential operators for a scale of Poisson type kernels in the unit disc. J. Anal. Math. 123, 227–249 (2014) Olofsson, A.: On a weighted harmonic Green function and a theorem of Littlewood. Bull. Sci. Math. 158, 102809 (2020) Olofsson, A.: Lipschitz continuity for weighted harmonic functions in the unit disc. Complex Var. Elliptic Equ. 65, 1630–1660 (2020) Olofsson, A., Wittsten, J.: Poisson integrals for standard weighted Laplacians in the unit disc. J. Math. Soc. Jpn. 65, 447–486 (2013) Wittsten, J.: Generalized axially symmetric potentials with distributional boundary values. Bull. Sci. Math. 139, 892–922 (2015)