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On the boundary terms in Hardy's inequalities for $W^{1,p}$ functions

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Ahmed A. Abdelhakim

Analysis of PDEs · Submitted 26 Dec 2017

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

The authors' main claim With the help of a radially invariant vector field, we derive inequalities of the Hardy kind, with no boundary terms, for $W^{1,p}$ functions on bounded star domains.

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arXiv ID1712.09173 (version 4)
Pith IDEPBOMYHU, Pith's permanent record of this paper and its review
CitationAhmed A. Abdelhakim. "On the boundary terms in Hardy's inequalities for $W^{1,p}$ functions." arXiv:1712.09173 (2017). https://pith.science/paper/EPBOMYHU
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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, 2,905 input tokens.

What the abstract itself states, as probabilities: claims first 0.12 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

With the help of a radially invariant vector field, we derive inequalities of the Hardy kind, with no boundary terms, for $W^{1,p}$ functions on bounded star domains. Our results are not obtainable from the classical inequalities for $W^{1,p}_{0}$ functions. Unlike in $W^{1,p}_{0}$, our inequalities admit maximizers that we describe explicitly.

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 With the help of a radially invariant vector field, we derive inequalities of the Hardy kind, with no boundary terms, for $W^{1,p}$ functions on bounded star domains. contribution 0.72 0.96 0.78 asserted 1.00
s1 Our results are not obtainable from the classical inequalities for $W^{1,p}_{0}$ functions. interpretation 0.36 0.67 0.13 asserted 1.00
s2 Unlike in $W^{1,p}_{0}$, our inequalities admit maximizers that we describe explicitly. result 0.97 0.93 0.22 asserted 1.00

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

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sources read
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  • relations to other papers
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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

With the help of a radially invariant vector field, we derive inequalities of the Hardy kind, with no boundary terms, for $W^{1,p}$ functions on bounded star domains. Our results are not obtainable from the classical inequalities for $W^{1,p}_{0}$ functions. Unlike in $W^{1,p}_{0}$, our inequalities admit maximizers that we describe explicitly.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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