Bernstein inequalities with nondoubling weights

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

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