{"paper":{"title":"The Helmholtz equation with $L^p$ data and Bochner-Riesz multipliers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Michael Goldberg","submitted_at":"2015-02-06T22:16:38Z","abstract_excerpt":"We prove the existence of $L^2$ solutions to the Helmholtz equation $(-\\Delta - 1)u = f$ in ${\\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\\mathbb R}^n)$ and satisfies the \"Fredholm condition\" that $\\hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(-\\Delta -1)u = -Vu + f$, are connected to the Limiting Absorption Principle for Schr\\\"odinger operators.\n  The same techniques are then used to prove that a wide range of $L^p \\mapsto L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.02066","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"}