Finite-infinite range inequalities in the complex plane

Abstract

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 c1, c2 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|ωnP|pdσ≤c1exp(−c2n)∫K|ωnP|pdσ. 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.