Home Mathematics Sharp estimates for the gradient of the generalized Poisson integral for a half-space
Article
Licensed
Unlicensed Requires Authentication

Sharp estimates for the gradient of the generalized Poisson integral for a half-space

  • EMAIL logo and
Published/Copyright: April 6, 2018
Become an author with De Gruyter Brill

Abstract

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function f on n is obtained under the assumption that f belongs to Lp. The explicit value of the coefficient is found for the cases p=1 and p=2.

MSC 2010: 31B10; 31C99

Dedicated to Vakhtang Kokilashvili on the occasion of his 80th birthday


References

[1] D. Khavinson, An extremal problem for harmonic functions in the ball, Canad. Math. Bull. 35 (1992), no. 2, 218–220. 10.4153/CMB-1992-031-8Search in Google Scholar

[2] G. Kresin, Sharp and maximized real-part estimates for derivatives of analytic functions in the disk, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 24 (2013), no. 1, 95–110. 10.4171/RLM/646Search in Google Scholar

[3] G. Kresin and V. Maz’ya, Sharp Real-Part Theorems. A Unified Approach, Lecture Notes in Math. 1903, Springer, Berlin, 2007. Search in Google Scholar

[4] G. Kresin and V. Maz’ya, Optimal estimates for the gradient of harmonic functions in the multidimensional half-space, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 425–440. 10.3934/dcds.2010.28.425Search in Google Scholar

[5] G. Kresin and V. Maz’ya, Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball, J. Math. Sci. (N.Y.) 169 (2010), no. 2, 167–187. 10.1007/s10958-010-0045-4Search in Google Scholar

[6] G. Kresin and V. Maz’ya, Sharp real-part theorems in the upper half-plane and similar estimates for harmonic functions, J. Math. Sci. (N.Y.) 179 (2011), no. 1, 144–163. 10.1007/s10958-011-0586-1Search in Google Scholar

[7] G. Kresin and V. Maz’ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, Math. Surveys Monogr. 183, American Mathematical Society, Providence, 2012. 10.1090/surv/183Search in Google Scholar

[8] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series. Vol. 1. Elementary Functions, “Nauka”, Moscow, 1986. Search in Google Scholar

Received: 2017-7-15
Accepted: 2018-12-15
Published Online: 2018-4-6
Published in Print: 2018-6-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 23.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2018-0026/html
Scroll to top button