Finite-infinite range inequalities in the complex plane

Let E ⫅ C be closed, ω be a suitable weight function on E , σ be a positive Borel measure on E . We discuss the conditions on ω and σ which ensure the existence of a fixed compact subset K of E with the following property. For any p , 0 < P ≤ ∞ , there exist positive constants c 1 , c 2 depending only on E , ω , σ and p such that for every integer n ≥ 1 and every polynomial P of degree at most n , ∫ E \ K | ω n P | p d σ ≤ c 1 exp ( − c 2 n ) ∫ K | ω n P | p d σ . In particular, we shall show that the support of a certain extremal measure is, in some sense, the smallest set K which works. The conditions on σ are formulated in terms of certain localized Christoffel functions related to σ .