A weak-type inequality of subharmonic functions

We prove the weak-type inequality $\lambda \mu(u+|v|\ge \lambda )\le(\alpha +2) \int _{\partial D}u\,d\mu$, $\lambda >0$, between a non-negative subharmonic function $u$ and an $\mathbb H$-valued smooth function $v$, defined on an open set containing the closure of a bounded domain $D$ in a Euclidean space $\mathbb R^n$, satisfying $|v(0)|\le u(0)$, $|\nabla v|\le|\nabla u|$ and $|\Delta v|\le \alpha \Delta u$, where $\alpha \ge 0$ is a constant. Here $\mu$ is the harmonic measure on $\partial D$ with respect to 0. This inequality extends Burkholder's inequality in which $\alpha =1$ and $\mathbb H=\mathbb R^\nu$, a Euclidean space.