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 R d . Our main result can be applied to a general class of (pseudo-)differential operators in R d of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian (−∆) s with s > 0 and, in particular, any polyharmonic operator (−∆) m with integer m 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 (R d ) in any dimension d 1. As a technical key result, we solve a phase retrieval problem for the Fourier transform in R d . To achieve this, we classify the case of equality in the corresponding Hardy-Littlewood majorant problem for the Fourier transform in R d . Applications: Radial Symmetry of OptimizersIn this section, we discuss some applications of Theorem 1 to show radial symmetry of optimizers. More precisely, we will consider the following examples.• Gagliardo-Nirenberg type inequalities with derivatives of arbitrary order.• Ground states for generalized NLS type (e. g. fourth-order NLS).

On symmetry of traveling solitary waves for dispersion generalized NLS

“…First, as the essential key point, we show that {ξ ∈ R n : | Qω,v(ξ)| > 0} is a connected set in R n . This fact then enables us to apply the recent rigidity result [17] for the Hardy-Littlewood majorant problem in R n to conclude the proof.…”

“…with some constants α ∈ R and β ∈ R n . In fact, such a "rigidity result" about the phase function (i. e. being just an affine function on R n ) can be deduce from the recent result in [17] on the Hardy-Littlewood majorant problem in R n , provided we know that the open set…”

“…Establishing this topological fact is the crux of this paper. We remark that in [17] where the symmetric-decreasing (Schwarz) symmetrization in R n was used, we always have that Ω is either an open ball or all of R n ; in particular, the set Ω is connected. However, for the Steiner symmetrization in n − 1 codimensions needed to define ♯e it is far from clear that the Ω is a connected set.…”

“…Let us assume p = 2m with some integer m ≥ 2 so that the corresponding dual exponent is given by p ′ = 2m 2m−1 . Since u ∈ L p (R n ) ∩ F(L p ′ (R n )), we can apply the version of the convolution lemma in [17] to conclude (3.9) u p L p = F(|u| 2m )(0) = ( u * u * · · · * u * u)(0), where the number of convolutions on the right-hand side equals 2m − 1. By Proposition 3.1, we obtain that ( u * u * · · · * u * u)(0) ≤ ( u * 1 * ( u) * 1 * · · · * ( u) * 1 * ( u) * 1 )(0) = F(|u ♯ 1 | 2m )(0) = u ♯ 1 p L p , where we also used the fact that F(u ♯ 1 ) = F(u) * 1 and the definition of ♯1.…”

Symmetry Results in Two-Dimensional Inequalities for Aharonov–Bohm Magnetic Fields

“…In such a context, standard techniques such as symmetrization have shown to be successful only in very particular situations, see e.g. [17] and [3,22]. To explain some of the ideas involved, we use polar coordinates and, when ψ does not vanish, writê…”