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Some Results for the Szeg\H{o} and Bergman Projections on Planar Domains

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Nathan A. Wagner

Complex Variables · Submitted 30 Aug 2022

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

The authors' main claim In particular, we prove weighted estimates for the projections, provide quantitative $L^p$ estimates and a specific example of such estimates on a domain with a sharp $p$ range, and show that the ``difference'' of the Bergman and Szeg\H{o} projections is compact at the endpoints $p = 1, \infty$ for domains with sufficient smoothness.

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arXiv ID2208.14512 (version 2)
Pith IDGI7SMMWM, Pith's permanent record of this paper and its review
CitationNathan A. Wagner. "Some Results for the Szeg\H{o} and Bergman Projections on Planar Domains." arXiv:2208.14512 (2022). https://pith.science/paper/GI7SMMWM
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,072 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.47 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.48 · compares with prior results 0.08 · mentions code or data availability 0.01.

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

The purpose of this note is to prove some boundedness/compactness results of a harmonic analysis flavor for the Bergman and Szeg\H{o} projections on certain classes of planar domains using conformal mappings. In particular, we prove weighted estimates for the projections, provide quantitative $L^p$ estimates and a specific example of such estimates on a domain with a sharp $p$ range, and show that the ``difference'' of the Bergman and Szeg\H{o} projections is compact at the endpoints $p = 1, \infty$ for domains with sufficient smoothness. We also pose some open questions that naturally arise from our investigation.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The purpose of this note is to prove some boundedness/compactness results of a harmonic analysis flavor for the Bergman and Szeg\H{o} projections on certain classes of planar domains using conformal mappings. contribution 0.98 0.63 0.27 asserted 0.95
s1 In particular, we prove weighted estimates for the projections, provide quantitative $L^p$ estimates and a specific example of such estimates on a domain with a sharp $p$ range, and show that the ``difference'' of the Bergman and Szeg\H{o} projections is compact at the endpoints $p = 1, \infty$ for domains with sufficient smoothness. result 0.65 0.97 0.74 asserted 1.00
s2 We also pose some open questions that naturally arise from our investigation. future work 0.99 0.72 0.02 asserted 0.78

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

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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

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Original abstract

The purpose of this note is to prove some boundedness/compactness results of a harmonic analysis flavor for the Bergman and Szeg\H{o} projections on certain classes of planar domains using conformal mappings. In particular, we prove weighted estimates for the projections, provide quantitative $L^p$ estimates and a specific example of such estimates on a domain with a sharp $p$ range, and show that the ``difference'' of the Bergman and Szeg\H{o} projections is compact at the endpoints $p = 1, \infty$ for domains with sufficient smoothness. We also pose some open questions that naturally arise from our investigation.

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