{"paper":{"title":"Extension technique for complete Bernstein functions of the Laplace operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Jacek Mucha, Mateusz Kwa\\'snicki","submitted_at":"2017-07-08T18:14:23Z","abstract_excerpt":"We discuss representation of certain functions of the Laplace operator $\\Delta$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-\\Delta)^{1/2}$, the square root of the $d$-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $(d + 1)$-dimensional Laplace operator $\\Delta_{t,x}$ in $(0, \\infty) \\times \\mathbf{R}^d$. Caffarelli and Silvestre extended this to fractional powers $(-\\Delta)^{\\alpha/2}$, which correspond to operators $\\nabla_{t,x} (t^{1 - \\alpha} \\nabla_{t,x})$. We provide an analogous result for all co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02475","kind":"arxiv","version":1},"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"}