Where does the $L^ p$-norm of a weighted polynomial live?

Abstract:

For a general class of nonnegative weight functions $w(x)$ having bounded or unbounded support $\Sigma \subset {\mathbf {R}}$, the authors have previously characterized the smallest compact set ${\mathfrak {S}_w}$, having the property that for every $n = 1, 2, \ldots$ and every polynomial $P$ of degree $\leqslant n$, \[ ||{[w(x)]^n}P(x)|{|_{{L^\infty }(\Sigma )}} = ||{[w(x)]^n}P(x)|{|_{{L^\infty }({\mathfrak {S}_w})}}\] . In the present paper we prove that, under mild conditions on $w$, the ${L^p}$-norms $(0 < p < \infty )$ of such weighted polynomials also "live" on ${\mathfrak {S}_w}$ in the sense that for each $\eta > 0$ there exist a compact set $\Delta$ with Lebesgue measure $m(\Delta ) < \eta$ and positive constants ${c_1}$, ${c_2}$ such that \[ ||{w^n}P|{|_{{L^p}(\Sigma )}} \leqslant (1 + {c_1}\exp ( - {c_2}n))||{w^n}P|{|_{{L^p}({\mathfrak {S}_w} \cup \Delta )}}\] . As applications we deduce asymptotic properties of certain extremal polynomials that include polynomials orthogonal with respect to a fixed weight over an unbounded interval. Our proofs utilize potential theoretic arguments along with Nikolskii-type inequalities.