Cauchy–Poisson transform and polynomial inequalities

Mirosław Baran¹

¹ Institute of Mathematics Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 95 (2009) no. 3, pp. 199-206.

Résumé

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 ${\mathbb{R}}^N$ 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 $E=[-1,1]\subset \mathbb{R}$.

Détail

DOI : 10.4064/ap95-3-1

Mots-clés : apply cauchy poisson transform prove multivariate polynomial inequalities particular pluricomplex green function fat compact set bbb lder continuous admits szeg type inequality weight function text dist partial kappa positive kappa viewed nontrivial generalization classical result interval subset bbb