Sharp weak-type inequalities for differentially subordinated martingales

Abstract

Let $M, N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities:

(i) If $M\geq0$ and $0<p\leq1$, then \[ \Vert N\Vert_{p,\infty} \leq2\Vert M\Vert_p \] and the constant is the best possible.

(ii) If $M\geq0$ and $p\geq2$, then \[ \Vert N\Vert_{p,\infty} \leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_p \] and the constant is the best possible.

(iii) If $1\leq p\leq2$ and $M$ and $N$ are orthogonal, then \[ \Vert N\Vert_{p,\infty} \leq K_p \Vert M\Vert_p, \] where \[ K_p^p=\frac{1}{\Gamma(p+1)}\cdot\biggl(\frac{\pi}{2}\biggr)^{p-1}\cdot\frac{1+1/3^2+ 1/5^2+1/7^2+\cdots}{ 1-1/3^{p+1}+1/5^{p+1}-1/7^{p+1}+\cdots}. \] The constant is the best possible.

We also provide related estimates for harmonic functions on Euclidean domains.