Bernstein inequalities with nondoubling weights

Abstract

We answer Totik's question on weighted Bernstein's inequalities showing that $$\Vert T_n^{\prime }{\Vert }_{L_p(\omega )}\le C(p,\omega )n\Vert T_n{\Vert }_{L_p(\omega )},0<p\le \infty ,$$ holds for all trigonometric polynomials $T_n$ and certain nondoubling weights $\omega$. Moreover, we find necessary conditions on $\omega$ for Bernstein's inequality to hold. We also prove weighted Bernstein-Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.

arXiv comment30 pages