Szegő's Condition on Compact subsets of $\mathbb{C}$

Abstract

Let $K$ be a non-polar compact subset of $\mathbb{C}$ and ${\mu }_K$ be its equilibrium measure. Let $\mu$ be a unit Borel measure supported on a compact set which contains the support of ${\mu }_K$. We prove that a Szegő condition in terms of the Radon-Nikodym derivative of $\mu$ with respect to ${\mu }_K$ implies that $$\underset{n}{\inf }\frac{\Vert P_n(\cdot ;\mu ){\Vert }_{L^2(\mathbb{C};\mu )}}{\mathrm{Cap}(K{)}^n}>0.$$ We show that $\frac{\Vert P_n(\cdot ;{\mu }_K){\Vert }_{L^2(\mathbb{C};{\mu }_K)}}{\mathrm{Cap}(K{)}^n}\ge 1$ for any compact non-polar set $K$. We also prove that under an additional assumption, unboundedness of the sequence $\left(\frac{\Vert P_n(\cdot ;{\mu }_K){\Vert }_{L^2(\mathbb{C};{\mu }_K)}}{\mathrm{Cap}(K{)}^n}\right)$ implies that $K$ satisfies the Parreau-Widom condition.