Three-term idempotent counterexamples in the Hardy–Littlewood majorant problem

Abstract

The Hardy-Littlewood majorant problem was raised in the 30's and it can be formulated as the question whether |f | p ≥ |g| p whenever f ≥ | g|. It has a positive answer only for exponents p which are even integers. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exists some finite set of exponentials and some ± signs with which the signed exponential sum has larger p th norm than the idempotent obtained with all the signs chosen + in the exponential sum. That conjecture was proved recently by Mockenhaupt and Schlag. However, a natural question is if even the classical 1+e 2πix ±e 2πi(k+2)x three-term exponential sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. We investigate the sharpened question and show that at least in certain cases there indeed exist three-term idempotent counterexamples in the Hardy-Littlewood majorant problem; that is we have for 0 < p < 6, p / ∈ 2NThe proof combines delicate calculus with numerical integration and precise error estimates. (2000): Primary 42A05. Mathematics Subject Classification

On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem

“…We have proved this for k = 0, 1, 2 in [8]. One motivation for us was the recent paper of Bonami and Révész [4], who used suitable idempotent polynomials as the base of their construction, via Riesz kernels, of highly concentrated ones in L p (T) for any p > 0.…”

“…In particular for [a, b] = [0, 9] we always have For the application of the above quadrature (11) we calculate (c.f. also [8])…”

“…Second, as already suggested in the conclusion of [8], we apply Taylor series expansion at more points than just at the midpoint t 0 := k + 1/2 of the t-interval (k, k + 1). This reduces the size of powers of (t − t 0 ), from powers of 1/2 to powers of smaller radii.…”

“…This reduces the size of powers of (t − t 0 ), from powers of 1/2 to powers of smaller radii. The Taylor polynomial of degree 7, considered in [8], had error size 2 −8 due to the contribution of |ξ − t 0 | 8 in the Lagrange remainder term, while here for k = 4 the division of the t-interval to (4, 4.5) and (4.5, 5) results in O(4 −n ) in the respective error contribution.…”

“…For details we refer to [4]. For the history and relevance of this closely related problem of idempotent polynomial concentration in L p see [4,5], the detailed introduction of [8], the survey paper [1], and the references therein.…”