Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

Volume 61 / 2013

Adam Osękowski Bulletin Polish Acad. Sci. Math. 61 (2013), 209-218 MSC: Primary 31B05; Secondary 60G44. DOI: 10.4064/ba61-3-3


Assume that $u$, $v$ are conjugate harmonic functions on the unit disc of $\mathbb{C}$, normalized so that $u(0)=v(0)=0$. Let $u^*$, $|v|^*$ stand for the one- and two-sided Brownian maxima of $u$ and $v$, respectively. The paper contains the proof of the sharp weak-type estimate $$ \mathbb{P}(|v|^*\geq 1)\leq \frac{1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\ldots}{1-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\ldots} \,\mathbb E u^*.$$ Actually, this estimate is shown to be true in the more general setting of differentially subordinate harmonic functions defined on Euclidean domains. The proof exploits a novel estimate for orthogonal martingales satisfying differential subordination.


  • Adam OsękowskiDepartment of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland

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