$L^p$-norms and Mahler's measure of polynomials on the $n$-dimensional torus

We prove Nikol’skii type inequalities which for polynomials on the $n$-dimensional torus ${\mathbb{T}}^n$ relate the $L^p$-with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0<p<q<\mathrm{\infty }$). Among other things we show that $C=\sqrt{q\mathrm{/}p}$ is the best constant such that $\mathrm{\parallel }P{\mathrm{\parallel }}_{L^q}\le C^{\text{deg}(P)}\mathrm{\parallel }P{\mathrm{\parallel }}_{L^p}$ for all homogeneous polynomials $P$ on ${\mathbb{T}}^n$. We also prove an exact inequality between the $L^p$-norm of a polynomial $P$ on ${\mathbb{T}}^n$ and its Mahler measure $M(P)$, which is the geometric mean of $\mathrm{\mid }P\mathrm{\mid }$ with respect to the normalized Lebesgue measure on ${\mathbb{T}}^n$. Using extrapolation we transfer this estimate into a Khintchine-Kahane type inequality, which, for polynomials on ${\mathbb{T}}^n$, relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.