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Abstract

In 1889 A. Markov proved that for every polynomial $p$ in one variable the inequality $\|p’\|_{[-1,1]}\leq (\deg p)^2 \|p\|_{[-1,1]}$ is true. Moreover, the exponent $2$ in this inequality is the best possible one. A tangential Markov inequality is a generalization of the Markov inequality to tangential derivatives of certain sets in higher-dimensional Euclidean spaces. We give some motivational examples of sets that admit the tangential Markov inequality with the sharp exponent. The main theorems show that the results on certain arcs and surfaces, which have been proved earlier for the uniform norm, can be generalized to $L^p$ norms.