Harmonic majorization and Brownian motion
Using probabilistic methods, we first give a very simple proof of the following recent theorem of \textit{M. Essén} [Ann. Acad. Sci. Fenn., Ser. A I 15, No. 1, 83-105 (1990; Zbl 0696.31008)]: Let \(D\) be a domain in \(\mathbb{R}^ n\), \(n \geq 2\). If \(p > 1\), then the subharmonic function \(x \to | x|^ p\) has a harmonic majorant in \(D\) if and only if the function \(x_ 1 \to | x_ 1|^ p\) has one, where \(x_ i\) denotes the \(i\)-th coordinate of \(x\). If \(p \leq 1\), the above equivalence is no longer true; in this case, we obtain a similar and new result, where \(| x_ 1|^ p\) is replaced by \((| x_ 1|^ 2 + | x_ 2|^ 2)^{p/2}\). Moreover, we answer some open question relative to the special case \(p = 1\), and obtain a new necessary and sufficient condition for \(x \to | x_ 1| \log^ + | x_ 1 |\) to have a harmonic majorant in \(D\). Our proofs involve classical relations between potential theory and Brownian motion, Burkholder-Gundy inequalities for martingales, Burkholder's results on harmonic majorization [\textit{D. L. Burkholder}, Adv. Math. 26, 182-205 (1977; Zbl 0372.60112)] and the authors' recent characterization of the class \(L \log L\) [\textit{J. Brossard} and \textit{L. Chevalier}, C. R. Acad. Sci., Paris, Ser. I 305, 135-137 (1987; Zbl 0624.60057)].
Burkholder- Gundy inequalities for martingales
scientific article; zbMATH DE number 561189
scientific article