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Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups

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Maochun Zhu, Marco Bramanti, Pengcheng Niu

Analysis of PDEs · Submitted 29 Mar 2012

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

The authors' main claim We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system.

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arXiv ID1203.6474 (version 2)
Pith IDS7XXUDCS, Pith's permanent record of this paper and its review
CitationMaochun Zhu, Marco Bramanti, Pengcheng Niu. "Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups." arXiv:1203.6474 (2012). https://pith.science/paper/S7XXUDCS
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,103 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.51 · compares with prior results 0.04 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system.

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 Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). background 1.00 0.07 0.01 asserted 0.97
s1 We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). background 0.51 0.38 0.03 asserted 0.92
s2 We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system. result 0.88 0.97 0.96 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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  • T body sentences
  • C citations
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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

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system.

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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

What it cites · 2 sentences without a confident label

Cites as background 2 papers
  • Meanwhile, regularity for linear degenerate elliptic equations involving vector fields has been investigated and many results have been proved, see for instance [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] and references therein; as for subelliptic systems structured on Hörmander's vector fields, we can quote [ref] , [ref] , [ref] .

    L^p and Schauder estimates for nonvariational operators structured on H\"ormander vector fields with drift · Introduction · 0.98

  • Meanwhile, regularity for linear degenerate elliptic equations involving vector fields has been investigated and many results have been proved, see for instance [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] and references therein; as for subelliptic systems structured on Hörmander's vector fields, we can quote [ref] , [ref] , [ref] .

    mathap/0502569 · Introduction · 0.98

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