Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

Adam Osękowski¹

¹ Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) no. 3, pp. 209-218

Résumé

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.

Détail

DOI : 10.4064/ba61-3-3

Keywords: assume conjugate harmonic functions unit disc mathbb normalized * * stand one two sided brownian maxima respectively paper contains proof sharp weak type estimate mathbb * geq leq frac frac frac frac ldots frac frac frac ldots mathbb * actually estimate shown general setting differentially subordinate harmonic functions defined euclidean domains proof exploits novel estimate orthogonal martingales satisfying differential subordination