Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

G. Kresin; V. Maz'ya

doi:10.1007/s10958-010-0045-4

Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

G. Kresin, V. Maz'ya

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L^p. This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L^p-norm of its boundary values is attained at the radial direction. This confirms D. Khavinson's conjecture for p = 1 and p = 2. Bibliography: 11 titles.

Original language	English
Pages (from-to)	167-187
Number of pages	21
Journal	Journal of Mathematical Sciences
Volume	169
Issue number	2
DOIs	https://doi.org/10.1007/s10958-010-0045-4
State	Published - 2010

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title = "Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball",

abstract = "A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to Lp. This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed Lp-norm of its boundary values is attained at the radial direction. This confirms D. Khavinson's conjecture for p = 1 and p = 2. Bibliography: 11 titles.",

author = "G. Kresin and V. Maz'ya",

note = "Funding Information: Acknowledgments. The research of the first author was supported by the KAMEA program of the Ministry of Absorption, State of Israel, and by the Ariel University Center of Samaria. The second author was partially supported by the UK Engineering and Physical Sciences Research Council grant EP/F005563/1.",

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N2 - A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to Lp. This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed Lp-norm of its boundary values is attained at the radial direction. This confirms D. Khavinson's conjecture for p = 1 and p = 2. Bibliography: 11 titles.

AB - A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to Lp. This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed Lp-norm of its boundary values is attained at the radial direction. This confirms D. Khavinson's conjecture for p = 1 and p = 2. Bibliography: 11 titles.

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Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

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