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Here we consider a kernel $K(x,y)=\\psi (y-a(x))+\\psi(x-a(y))$ where $\\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on $\\rr^d$. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\\psi$. Indeed, in the linear case $a(x) = Ax$ we obtain an explicit expression for the first eigenvalue in the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.4114","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-17T14:44:13Z","cross_cats_sorted":[],"title_canon_sha256":"95919cfc62ce472b737a5e2e8d6cb5d6b5972f021f1ecf6422273f008cfb79e1","abstract_canon_sha256":"3d6c7fc0ad9a0db861dffefeb0c7ae57d2e6436f4739315d03a05ac0faf7095d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:05.663763Z","signature_b64":"EA7s1rh6xBDn/IKynCHJ0HFjAQIRyrlU7FzFieYQXX069Kp81YzkhcG68q5BU9V9fcMbT2EgZAp9eKxvPxPcBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aae12d523d6a7e3f996297e44d69b2978668d75817dd55377f84400b65eaf393","last_reissued_at":"2026-05-18T04:08:05.663259Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:05.663259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. San Antolin, J. D. Rossi, L. I. Ignat","submitted_at":"2011-11-17T14:44:13Z","abstract_excerpt":"We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \\int_{\\rr^d} K(x,y) (u(y)-u(x)) \\, dy$. Here we consider a kernel $K(x,y)=\\psi (y-a(x))+\\psi(x-a(y))$ where $\\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on $\\rr^d$. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\\psi$. 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