Notes on Inequalities with Doubling Weights

Various 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, like in the weighted Nikolskii inequality, the slightly stronger A∞ condition is used. Throughout their paper the Lp norm 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 the pth power (0 < p < ∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases. 1. The Weights For Introduction we refer to Sections 1 and 2 of the Mastroianni-Totik paper [12] and the references therein. See [1] – [9], [11], and [13]. Here we just formulate the original and some equivalent definitions that we shall use. In Sections 2 – 7 we shall work with integrable, 2π-periodic weight functions W satisfying the so-called doubling condition: (1.1) W (2I) ≤ LW (I) for intervals I ⊂ R, where L is a constant independent of I, 2I is the interval with length 2|I| (|I| denotes the length of the interval I) and with midpoint at the midpoint of I, and W (I) := ∫ I W (u) du . In other words, W has the doubling property if the measure of a twice enlarged interval is less than a constant times the measure of the original interval. An integrable, 2π periodic weight function on R satisfying the doubling condition will be called a doubling weight. We start with the following elementary observation. 1991 Mathematics Subject Classification. Primary: 41A17.