Lp Markov-Bernstein Inequalities for Freud Weights Original Research Article

Let W(x) ≔ exp(−Q(x)), x ∈ R, where Q(x) is even and continuous in R, Q(0) = 0 and Q″ is continuous in (0, ∞) with Q′(x) > 0 in (0, ∞), and for some A, B > 1, A ≤ (xQ′(x))′/Q′(x) ≤ B, x ∈ (0, ∞). For example, Q(x)≔|x|α, α > 1 satisfies these hypotheses. Let an denote the nth Mhaskar-Rahmanov-Saff number for Q, and [formula]. Let 1 ≤ p < ∞. We prove that for n ≥ 1 and polynomials P of degree at most n, [formula]. This extends to Lp the recent L∞ result of the authors, in which the essential feature is the introduction of the factor ψ−12n. We also consider the case A ≤ 1. The proofs are necessarily different from previous methods of extending L∞ inequalities to Lp and involve Carleson measures.