{"paper":{"title":"The Fourier transform on negatively curved harmonic manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kingshook Biswas","submitted_at":"2018-02-20T18:30:25Z","abstract_excerpt":"Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \\leq -1$, and let $\\partial X$ denote the boundary at infinity of $X$. Let $h > 0$ denote the mean curvature of horospheres in $X$, and let $\\rho = h/2$. Fixing a basepoint $o \\in X$, for $\\xi \\in \\partial X$, let $B_{\\xi}$ denote the Busemann function at $\\xi$ such that $B_{\\xi}(o) = 0$, then for $\\lambda \\in \\mathbb{C}$ the function $e^{(i\\lambda - \\rho)B_{\\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\\lambda^2 + \\rho^2)$.\n  For a function $f$ on $X$, we defi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.07236","kind":"arxiv","version":2},"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"}