Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The solution operator of the inhomogeneous Dirichlet problem in the unit ball


Authors: David Kalaj and Djordjije Vujadinović
Journal: Proc. Amer. Math. Soc. 144 (2016), 623-635
MSC (2010): Primary 35J05; Secondary 47G10
DOI: https://doi.org/10.1090/proc/12723
Published electronically: August 26, 2015
MathSciNet review: 3430840
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we estimate norms of integral operator induced by the Green function related to the Poisson equation in the unit ball with vanishing boundary data.


References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. MR 725161 (84m:30028)
  • [2] J. M. Anderson and A. Hinkkanen, The Cauchy transform on bounded domains, Proc. Amer. Math. Soc. 107 (1989), no. 1, 179-185. MR 972226 (90c:30065), https://doi.org/10.2307/2048052
  • [3] J. M. Anderson, D. Khavinson, and V. Lomonosov, Spectral properties of some integral operators arising in potential theory, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 172, 387-407. MR 1188382 (93j:47073), https://doi.org/10.1093/qmathj/43.4.387
  • [4] J. Arazy and D. Khavinson, Spectral estimates of Cauchy's transform in $ L^2(\Omega )$, Integral Equations Operator Theory 15 (1992), no. 6, 901-919. MR 1188786 (93k:47058), https://doi.org/10.1007/BF01203120
  • [5] Milutin R. Dostanić, Norm estimate of the Cauchy transform on $ L^p(\Omega )$, Integral Equations Operator Theory 52 (2005), no. 4, 465-475. MR 2184599 (2006i:42015), https://doi.org/10.1007/s00020-002-1290-9
  • [6] Milutin R. Dostanić, Estimate of the second term in the spectral asymptotic of Cauchy transform, J. Funct. Anal. 249 (2007), no. 1, 55-74. MR 2338854 (2009b:30077), https://doi.org/10.1016/j.jfa.2007.04.007
  • [7] Milutin R. Dostanić, The properties of the Cauchy transform on a bounded domain, J. Operator Theory 36 (1996), no. 2, 233-247. MR 1432117 (98m:47041)
  • [8] S. S. Dragomir, R. P. Agarwal, and N. S. Barnett, Inequalities for beta and gamma functions via some classical and new integral inequalities, J. Inequal. Appl. 5 (2000), no. 2, 103-165. MR 1753533 (2000m:26012), https://doi.org/10.1155/S1025583400000084
  • [9] Håkan Hedenmalm and Serguei Shimorin, Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J. 127 (2005), no. 2, 341-393. MR 2130416 (2005m:30010), https://doi.org/10.1215/S0012-7094-04-12725-3
  • [10] Ricardo G. Durán, Marcela Sanmartino, and Marisa Toschi, Weighted a priori estimates for the Poisson equation, Indiana Univ. Math. J. 57 (2008), no. 7, 3463-3478. MR 2492240 (2010e:35058), https://doi.org/10.1512/iumj.2008.57.3427
  • [11] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [12] David Kalaj, On some integral operators related to the Poisson equation, Integral Equations Operator Theory 72 (2012), no. 4, 563-575. MR 2904611, https://doi.org/10.1007/s00020-012-1952-1
  • [13] David Kalaj, Cauchy transform and Poisson's equation, Adv. Math. 231 (2012), no. 1, 213-242. MR 2935387, https://doi.org/10.1016/j.aim.2012.05.003
  • [14] David Kalaj and Miroslav Pavlović, On quasiconformal self-mappings of the unit disk satisfying Poisson's equation, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4043-4061. MR 2792979 (2012b:30045), https://doi.org/10.1090/S0002-9947-2011-05081-6
  • [15] Dmitry Khavinson, On uniform approximation by harmonic functions, Michigan Math. J. 34 (1987), no. 3, 465-473. MR 911819 (89a:41026), https://doi.org/10.1307/mmj/1029003626
  • [16] G. O. Thorin, Convexity theorems generalizing those of M. Riesz and Hadamard with some applications, Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 9 (1948), 1-58. MR 0025529 (10,21e)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J05, 47G10

Retrieve articles in all journals with MSC (2010): 35J05, 47G10


Additional Information

David Kalaj
Affiliation: Faculty of Mathematics, University of Montenegro, Dzordza Vašingtona bb, 81000 Podgorica, Montenegro

Djordjije Vujadinović
Affiliation: Faculty of Mathematics, University of Montenegro, Dzordza Vašingtona bb, 81000 Podgorica, Montenegro
Email: djordjijevuj@t-com.me

DOI: https://doi.org/10.1090/proc/12723
Keywords: M\"obius transformations, Poisson equation, Newtonian potential, Cauchy transform, Bessel function
Received by editor(s): October 9, 2014
Received by editor(s) in revised form: January 12, 2015
Published electronically: August 26, 2015
Communicated by: Franc Forstneric
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society