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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A weak-type inequality for
differentially subordinate harmonic functions


Author: Changsun Choi
Journal: Trans. Amer. Math. Soc. 350 (1998), 2687-2696
MSC (1991): Primary 31B05, 31B15; Secondary 42A50, 60G42
MathSciNet review: 1617340
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Abstract: Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder $\mu(|v|\ge 1)\le 2\|u\|_1$ for two harmonic functions $u$ and $v$. That is, we prove the sharp weak-type inequality $\mu(|v|\ge 1)\le K\|u\|_1$ under the assumptions that $|v(\xi)|\le |u(\xi)|$, $|\nabla v|\le|\nabla u|$ and the extra assumption that $\nabla u\cdot\nabla v=0$. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant $K$ is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.


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Additional Information

Changsun Choi
Affiliation: Department of Mathematics, University of Illinois, 273 Altgeld Hall, 1409 West Green Street, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, KAIST Taejon, 305-701 Korea
Email: cschoi@math.kaist.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02259-4
PII: S 0002-9947(98)02259-4
Keywords: Harmonic functions, harmonic measure, differential subordination, weak-type inequality, Burkholder's inequality, Kolmogorov's inequality, Davis's constant
Received by editor(s): October 7, 1995
Article copyright: © Copyright 1998 American Mathematical Society