{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5FO4AMD6DQXI2YYOVC7BTL6PA3","short_pith_number":"pith:5FO4AMD6","canonical_record":{"source":{"id":"1611.10304","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T18:45:47Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"d3f2703ee8b8398a0f8c268b2c21c9a44e15bcb63a8fb956512f691870944027","abstract_canon_sha256":"92e920fc637f70f0baa4730477413e50c14cbf21272e71c9f754e2f3e2088b09"},"schema_version":"1.0"},"canonical_sha256":"e95dc0307e1c2e8d630ea8be19afcf06c19f662de12710ba485c0a09a9978882","source":{"kind":"arxiv","id":"1611.10304","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.10304","created_at":"2026-05-18T00:43:58Z"},{"alias_kind":"arxiv_version","alias_value":"1611.10304v2","created_at":"2026-05-18T00:43:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.10304","created_at":"2026-05-18T00:43:58Z"},{"alias_kind":"pith_short_12","alias_value":"5FO4AMD6DQXI","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5FO4AMD6DQXI2YYO","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5FO4AMD6","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5FO4AMD6DQXI2YYOVC7BTL6PA3","target":"record","payload":{"canonical_record":{"source":{"id":"1611.10304","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T18:45:47Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"d3f2703ee8b8398a0f8c268b2c21c9a44e15bcb63a8fb956512f691870944027","abstract_canon_sha256":"92e920fc637f70f0baa4730477413e50c14cbf21272e71c9f754e2f3e2088b09"},"schema_version":"1.0"},"canonical_sha256":"e95dc0307e1c2e8d630ea8be19afcf06c19f662de12710ba485c0a09a9978882","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:58.696729Z","signature_b64":"JNPnuV0XFnV0U+BfU+8GGSLninFfSMBURRuW6EJdp3VUh/lsyo4sOE1iuGhrFm1bp8isEcblin8b+h21sutZDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e95dc0307e1c2e8d630ea8be19afcf06c19f662de12710ba485c0a09a9978882","last_reissued_at":"2026-05-18T00:43:58.696159Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:58.696159Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.10304","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:43:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NkFRbG9ViYgATCOV2pgIlyjIxK8GTyX56ElO4no10V9qWbicLdqv5YUfR2g0IS4NFwCFeQgZsfAfAckJgFFKBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T10:26:36.120256Z"},"content_sha256":"81f681473df8bb03fc0814ff473b0e26b6855a9232652830fb485badfdf8c297","schema_version":"1.0","event_id":"sha256:81f681473df8bb03fc0814ff473b0e26b6855a9232652830fb485badfdf8c297"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5FO4AMD6DQXI2YYOVC7BTL6PA3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal L\\'evy processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Mateusz Kwa\\'snicki, Tomasz Grzywny","submitted_at":"2016-11-30T18:45:47Z","abstract_excerpt":"In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal L\\'evy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the L\\'{e}vy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10304","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:43:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ewU1gd/CPob/FPWowZyU5voGlJRpM5ZGTiZmjC7KJ1gj/GlGCmOOrN3JSFzwwsfDlNmY9e17kuDN0p58+rkbAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T10:26:36.120822Z"},"content_sha256":"9a8c60c2c7ace205ea6eb0c0c7d2ec1474c661027490d916b2c5720a8dac046e","schema_version":"1.0","event_id":"sha256:9a8c60c2c7ace205ea6eb0c0c7d2ec1474c661027490d916b2c5720a8dac046e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5FO4AMD6DQXI2YYOVC7BTL6PA3/bundle.json","state_url":"https://pith.science/pith/5FO4AMD6DQXI2YYOVC7BTL6PA3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5FO4AMD6DQXI2YYOVC7BTL6PA3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T10:26:36Z","links":{"resolver":"https://pith.science/pith/5FO4AMD6DQXI2YYOVC7BTL6PA3","bundle":"https://pith.science/pith/5FO4AMD6DQXI2YYOVC7BTL6PA3/bundle.json","state":"https://pith.science/pith/5FO4AMD6DQXI2YYOVC7BTL6PA3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5FO4AMD6DQXI2YYOVC7BTL6PA3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5FO4AMD6DQXI2YYOVC7BTL6PA3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"92e920fc637f70f0baa4730477413e50c14cbf21272e71c9f754e2f3e2088b09","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T18:45:47Z","title_canon_sha256":"d3f2703ee8b8398a0f8c268b2c21c9a44e15bcb63a8fb956512f691870944027"},"schema_version":"1.0","source":{"id":"1611.10304","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.10304","created_at":"2026-05-18T00:43:58Z"},{"alias_kind":"arxiv_version","alias_value":"1611.10304v2","created_at":"2026-05-18T00:43:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.10304","created_at":"2026-05-18T00:43:58Z"},{"alias_kind":"pith_short_12","alias_value":"5FO4AMD6DQXI","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5FO4AMD6DQXI2YYO","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5FO4AMD6","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:9a8c60c2c7ace205ea6eb0c0c7d2ec1474c661027490d916b2c5720a8dac046e","target":"graph","created_at":"2026-05-18T00:43:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal L\\'evy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the L\\'{e}vy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with r","authors_text":"Mateusz Kwa\\'snicki, Tomasz Grzywny","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T18:45:47Z","title":"Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal L\\'evy processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10304","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:81f681473df8bb03fc0814ff473b0e26b6855a9232652830fb485badfdf8c297","target":"record","created_at":"2026-05-18T00:43:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"92e920fc637f70f0baa4730477413e50c14cbf21272e71c9f754e2f3e2088b09","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T18:45:47Z","title_canon_sha256":"d3f2703ee8b8398a0f8c268b2c21c9a44e15bcb63a8fb956512f691870944027"},"schema_version":"1.0","source":{"id":"1611.10304","kind":"arxiv","version":2}},"canonical_sha256":"e95dc0307e1c2e8d630ea8be19afcf06c19f662de12710ba485c0a09a9978882","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e95dc0307e1c2e8d630ea8be19afcf06c19f662de12710ba485c0a09a9978882","first_computed_at":"2026-05-18T00:43:58.696159Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:58.696159Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JNPnuV0XFnV0U+BfU+8GGSLninFfSMBURRuW6EJdp3VUh/lsyo4sOE1iuGhrFm1bp8isEcblin8b+h21sutZDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:58.696729Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.10304","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:81f681473df8bb03fc0814ff473b0e26b6855a9232652830fb485badfdf8c297","sha256:9a8c60c2c7ace205ea6eb0c0c7d2ec1474c661027490d916b2c5720a8dac046e"],"state_sha256":"b844cc8bd3d2122d88cd41c6bdc560ffd0227860d5822a0ccfcf90c1893296fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HPKVHRsCnwoEqqhwl7j+etMQJOgSmyV9IgYM46fNwgmnxuqO5OPF44pCeBiVMq6IlY3eaVT2YuSoisVwgZptCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T10:26:36.123704Z","bundle_sha256":"394b7d1acdae3b342ce6f8d68d1ae7f28893e65c37006639668c533922ea9216"}}