Lp Markov–Bernstein Inequalities on All Arcs of the Circle

Abstract

Let 0<p<∞ and 0⩽α<β⩽2π. We prove that for n⩾1 and trigonometric polynomials sn of degree ⩽n, we have∫βα|s′n(θ)|psinθ−α2sinθ−β2+β−αn2cosθ−α+β222+1n2p/2dθ⩽cnp∫βα|sn(θ)|pdθ, where c is independent of α, β, n, sn. The essential feature is the uniformity in [α,β] of the estimate and the fact that as [α,β] approaches [0,2π], we recover the Lp Markov inequality. The result may be viewed as the complete Lp form of Videnskii's inequalities, improving earlier work of the second author.