{"paper":{"title":"The structure of solution spaces for fractional-order operators, with gradient estimates","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Gerd Grubb","submitted_at":"2026-07-02T15:24:53Z","abstract_excerpt":"The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-\\Delta )^{a}$ ($0<a<1$) or a pseudodifferential generalization $P$, on a bounded open set $\\Omega \\subset R^n$ with $C^{1+\\tau }$-boundary, $$ Pu=f \\text{ on }\\Omega ,\\quad u=0 \\text{ on }R^n\\setminus \\Omega , $$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in H\\\"older-Zygmund spaces $C_*^t$, that the solution space for $f$ of regularity $s\\in [0,\\tau -2a)$ is the direct sum of a component $\\dot H_q^{2a+s}(\\bar\\Omega)$ resp. $\\dot C_*^{2a+s}(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02312","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02312/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}