A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order

Abstract

We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in ${\mathbb{R}}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in ${\mathbb{R}}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-\Delta {)}^s$ with $s>0$ and, in particular, any polyharmonic operator $(-\Delta {)}^m$ with integer $m\ge 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of arbitrary order, ii) ground states for bi- and polyharmonic NLS, and iii) Adams-Moser-Trudinger type inequalities for $H^{d/2}({\mathbb{R}}^d)$ in any dimension $d\ge 1$. As a technical key result, we solve a phase retrieval problem for the Fourier transform in ${\mathbb{R}}^d$. To achieve this, we classify the case of equality in the corresponding Hardy-Littlewood majorant problem for the Fourier transform in ${\mathbb{R}}^d$.

25 pages. Second revised version. Comments are welcome