A weak-type inequality for differentially subordinate harmonic functions

Abstract

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.