Lᵖ-norms and Mahler's measure of polynomials on the n-dimensional torus

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We prove Nikol'skii type inequalities which for polynomials on the n-dimensional torus Tⁿ relate the Lᵖ-with the Lq-norm (with respect to the normalized Lebesgue measure and 0 <p <q < ∞). Among other things we show that C=√q/p is the best constant such that Pₗq≤ Cdeg(P) Pₗₚ for all homogeneous polynomials P on Tⁿ. We also prove an exact inequality between the Lᵖ-norm of a polynomial P on Tⁿ and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on Tⁿ. Using extrapolation we transfer this estimate into a Khintchine-Kahane type inequality, which, for polynomials on Tⁿ, relates a certain exponential Orlicz norm and Mahler's measure. Applications are given, including some interpolation estimates.