Plemelj–Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators
Bibliographic record
Abstract
Abstract Let R be a compact Riemann surface and $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> be a Jordan curve separating R into connected components $$\Sigma _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and $$\Sigma _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . We consider Calderón–Zygmund type operators $$T(\Sigma _1,\Sigma _k)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> taking the space of $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> anti-holomorphic one-forms on $$\Sigma _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> to the space of $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> holomorphic one-forms on $$\Sigma _k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> for $$k=1,2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> anti-holomorphic one-forms on R with respect to the inner product on $$\Sigma _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . We show that the restriction of the Schiffer operator $$T(\Sigma _1,\Sigma _2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to V is an isomorphism onto the set of exact holomorphic one-forms on $$\Sigma _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . Using the relation between this Schiffer operator and a Cauchy-type integral involving Green’s function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".