Lp Markov-Bernstein Inequalities for Freud Weights

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|�, � > 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 ��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.

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