| 研究実績の概要 |
The main focus this year has been on establishing one of the key aims of this research project - to prove, in full generality, the nonlinear Brascamp-Lieb conjecture. First, in collaboration with Jonathan Bennett, Stefan Buschenhenke and Taryn Flock, we were able to establish the conjecture for so-called simple Brascamp-Lieb data using a proof based on an induction-on-scales argument. Following this, in further collaborative work with the above researchers and Michael Cowling, we successfully established the nonlinear Brascamp-Lieb conjecture in full generality. In addition to an induction-on-scales argument, our argument makes use of a certain quantitative version of a famous theorem of Lieb concerning the extremisability of the classical Brascamp-Lieb inequality by gaussian functions. In order to establish such a quantitative version of Lieb’s theorem, we were required us to answer a natural question arising in optimisation theory concerning the minimisers and near-minimisers of weighted sums of exponential functions in arbitrary dimensions. As an application of our nonlinear Brascamp-Lieb inequality, we established estimates for multiple convolutions of Lebesgue space densities supported on submanifolds satisfying a certain transversality condition.
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