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Estimates and structure of $\alpha$-harmonic functions

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Krzysztof Bogdan, Tadeusz Kulczycki, Mateusz Kwaśnicki

Probability · Submitted 22 Jul 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$.

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arXiv IDmath/0607561 (version 2)
Pith IDMBFW7VF6, Pith's permanent record of this paper and its review
CitationKrzysztof Bogdan, Mateusz Kwa\'snicki, Tadeusz Kulczycki. "Estimates and structure of $\alpha$-harmonic functions." arXiv:math/0607561 (2006). https://pith.science/paper/MBFW7VF6
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,910 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. This yields a unique representation of such functions as integrals against measures on $D^c\cup \{\infty\}$ satisfying an integrability condition. The corresponding Martin boundary of $D$ is a subset of the Euclidean boundary determined by an integral test.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. result 0.66 0.96 0.74 asserted 1.00
s1 This yields a unique representation of such functions as integrals against measures on $D^c\cup \{\infty\}$ satisfying an integrability condition. result 0.88 0.90 0.25 asserted 1.00
s2 The corresponding Martin boundary of $D$ is a subset of the Euclidean boundary determined by an integral test. result 0.95 0.82 0.18 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. This yields a unique representation of such functions as integrals against measures on $D^c\cup \{\infty\}$ satisfying an integrability condition. The corresponding Martin boundary of $D$ is a subset of the Euclidean boundary determined by an integral test.

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