Orthogonal Polynomials Associated with Equilibrium Measures on $\mathbb {R}$

Let K be a non-polar compact subset of \(\mathbb {R}\) and μ _K denote the equilibrium measure of K. Furthermore, let P _n (?;μ _K ) be the n-th monic orthogonal polynomial for μ _K . It is shown that \(\|P_{n}\left (\cdot ; \mu _{K}\right )\|_{L^{2}(\mu _{K})}\), the Hilbert norm of P _n (?;μ _K ) in L ²(μ _K ), is bounded below by Cap(K) ⁿ for each \(n\in \mathbb {N}\). A sufficient condition is given for\(\left (\|P_{n}\left (\cdot ;\mu _{K}\right )\|_{L^{2}(\mu _{K})}/\text {Cap}(K)^{n}\right )_{n=1}^{\infty }\) to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.