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.