A weak-type inequality for differentially subordinate harmonic functions

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Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder mu(\v\ greater than or equal to 1) less than or equal to 2\\u\\(1) for two harmonic functions u and v. That is, we prove the sharp weak-type inequality mu(\v\ greater than or equal to 1) less than or equal to K\\u\\(1) under the assumptions that \v(xi)\ less than or equal to \u(xi)\, \del v\ less than or equal to \del u\ and the extra assumption that del u . del 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.
Publisher
Amer Mathematical Soc
Issue Date
1998-07
Language
English
Article Type
Article
Keywords

SHARP INEQUALITIES; MARTINGALES

Citation

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.350, no.7, pp.2687 - 2696

ISSN
0002-9947
URI
http://hdl.handle.net/10203/73067
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