Orthogonal polynomials associated with equilibrium measures on $\mathbb{R}$

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