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 (nin 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.