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

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 $m \geq 1$. 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 $d \geq 1$. 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 $\mathbb{R}^d$.