Special quadrature error estimates and their application in the hardy-littlewood majorant problem

Abstract

The Hardy-Littlewood majorant problem has a positive answer only for expo- nents p which are even integers, while there are counterexamples for all p =2 2N. Montgomery conjectured that there exist counterexamples even among idempotent polynomials. This was proved recently by Mockenhaupt and Schlag with some four-term idempotents. However, Mockenhaupt conjectured that even the classical 1 + e^{2\piix} \pm e^{2\pii(k+2)x} three- term character sums, should work for all 2k < p < 2k+2 and for all k \in N. In two previous papers we proved this conjecture for k = 0; 1; 2; 3; 4, i.e. in the range 0 < p < 10, p \notin 2N. Here we demonstrate that even the k = 5 case holds true. Refinements in the technical features of our approach include use of total variation and integral mean estimates in error bounds for a certain fourth order quadrature. Our estimates make good use of the special forms of functions we encounter: linear combinations of powers and powers of logarithms of absolute value squares of trigonometric polynomials of given degree. Thus the quadrature error estimates are less general, but we can find better constants which are of practical use for us.