{"paper":{"title":"A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.FA","math.MP"],"primary_cat":"math.AP","authors_text":"Enno Lenzmann, J\\'er\\'emy Sok","submitted_at":"2018-05-16T13:20:32Z","abstract_excerpt":"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 deri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06294","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}