Notes on Inequalities with Doubling Weights

AbstractVarious important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Sometimes, however, as in the weighted Nikolskii inequality, the slightly strongerA∞condition is used. Throughout their paper theLpnorm is studied under the assumption 1⩽p<∞. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0<p<1. The crucial tool is an estimate for quadrature sums for thepth power (0<p<∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases