A weak-type inequality for differentially subordinate harmonic functions

Abstract:

Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder $\mu (|v|\ge 1)\le 2\|u\|_1$ 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.