Cauchy–Poisson transform and polynomial inequalities

Volume 95 / 2009

Mirosław Baran Annales Polonici Mathematici 95 (2009), 199-206 MSC: 31C10, 32U35, 41A17. DOI: 10.4064/ap95-3-1


We apply the Cauchy–Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set $E$ in $\Bbb R^N$ is Hölder continuous then $E$ admits a Szegö type inequality with weight function $\text{dist}(x,\partial E)^{-(1-\kappa )}$ with a positive $\kappa$. This can be viewed as a (nontrivial) generalization of the classical result for the interval $E=[-1,1]\subset \Bbb R$.


  • Mirosław BaranInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland

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