Effective multipliers for weights whose log are H\"older continuous. Application to the cost of fast boundary controls for the 1D Schr{\"o}dinger equation

Abstract

We give a simple proof of the Beurling-Malliavin multiplier theorem (BM1) in the particular case of weights that verify the usual finite logarithmic integral condition and such that their log are H{\"o}lder continuous with exponent less than 1. Our proof has the advantage to give an explicit version of BM1, in the sense that one can give precise estimates from below and above for the multiplier, in terms of the exponential type we want to reach, and the constants appearing in the H{\"o}lder condition of our weights. The same ideas can be applied to a particular weight, that will lead to an improvement on the estimation of the cost of fast boundary controls for the 1D Schr{\"o}dinger equation on a segment. Our proof is mainly based on the use of a modified Hilbert transform together with its link with the harmonic extension in the complex upper half plane and some modified conjugate harmonic extension in the upper half plane.