$$L^{p}$$ L p -Norms and Mahler’s Measure of Polynomials on the n-Dimensional Torus

Andreas Defant,Mieczysław Mastyło

doi:10.1007/s00365-015-9319-x

Abstract

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus $\mathbb {T}^n$, relate the $L^p$-norm with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0 <p <q < \infty $). Among other things, we show that $C=\sqrt{q/p}$ is the best constant such that $\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}$ 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 |P| 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.

Highlights

The Mahler measure of a polynomial P on Cn is given by 1/ pM(P) := exp log |P| dz = lim |P|p dz, (1)Tn p→0+ Tn where dz = dz1 . . . dzn stands for the normalized Lebesgue measure on the n-torus Tn
We prove an exact inequality between the L p-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
We point out that Mahler [14] used the functional M as a powerful tool in a simple proof of the “Gelfond–Mahler inequality,” which found important applications in transcendence theory

Summary

Introduction

The following extension of (4), a sharp Khintchine–Kahane type inequality that allows one to estimate the L p(Tn)-norm P L p(Tn) of P ∈ P(Cn) by its Mahler measure M(P), was the starting point of this article, and will be proved in Sect. In [4, Theorem 9] Bayart used a standard iteration argument through Fubini’s theorem in order to extend (10) to polynomials in several variables: The constant r = This Khintchine–Kahane type inequality extends earlier work of Beauzamy et al from [5] ( p = 2 < q and p < q = 2) and Bonami [7] (q = 2, p = 1). Several interesting applications are in order—some motivated by the original work of Mahler and his followers

L p-Norms Versus Lq-Norms of Polynomials on Tn

L p-Norms Versus Mahler’s Measure for Polynomials on Tn

Applications