TY - JOUR

T1 - Sharp real-part theorems in the upper half-plane and similar estimates for harmonic functions

AU - Kresin, G.

AU - Maz'ya, V.

N1 - Funding Information:
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 Scandinavian Network “Analysis and Applications.”

PY - 2011/11

Y1 - 2011/11

N2 - Explicit formulas for sharp coefficients in estimates of the modulus of an analytic function and its derivative in the upper half-plane are found. It is assumed that the boundary values of the real part of the function are in Lp. As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in ℝ2+ are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in ℝn+ by the Lp-norm of the boundary normal derivative is given, 1 ≤ p < ∞. This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p ∈ [1, n]. Bibliography: 6 titles.

AB - Explicit formulas for sharp coefficients in estimates of the modulus of an analytic function and its derivative in the upper half-plane are found. It is assumed that the boundary values of the real part of the function are in Lp. As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in ℝ2+ are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in ℝn+ by the Lp-norm of the boundary normal derivative is given, 1 ≤ p < ∞. This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p ∈ [1, n]. Bibliography: 6 titles.

UR - http://www.scopus.com/inward/record.url?scp=80255138620&partnerID=8YFLogxK

U2 - 10.1007/s10958-011-0586-1

DO - 10.1007/s10958-011-0586-1

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AN - SCOPUS:80255138620

SN - 1072-3374

VL - 179

SP - 144

EP - 163

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

IS - 1

ER -