Weighted Polynomial Inequalities with Doubling and A∞ Weights

We consider weighted inequalities such as Bernstein, Nikolskii, Remez, etc., inequalities under minimal assumptions on the weights. It turns out that in most cases this mimimal assumption is the doubling condition. Sometimes, however, as for the Remez and Nikolskii inequalities, one needs the slightly stronger A∈fty condition. We shall consider both the trigonometric and the algebraic cases.