One-Parameter Semigroups Generated by Strongly M-Elliptic Pseudo-Differential Operators on Euclidean Spaces
Bibliographic record
Abstract
We begin with a recall of the definitions and basic properties of the standard Hörmander classes of pseudo-differential operators on Rn. Then we introduce a new class of pseudo-differential operators that can be traced back to Taylor, generalized by Garello and Morando and further developed by M. W. Wong. A related class of pseudo-differential operators depending on a complex parameter on an open subset of the complex plane is constructed. We tease out from this related class the strongly M – elliptic pseudo-differential operators and prove that they are infinitesimal generators of holomorphic and hence strongly continuous one-parameter semigroups of bounded linear operators on Lp(Rn), 1<p<ꚙ. The proof is based on careful refinements of the Agmon – Douglis – Nirenberg estimates for the pseudo-differential operators in the book by M. W. Wong. In the case when p=2, we give another proof that strongly (ρ,Λ) – elliptic pseudo-differential operators, which include strongly M – elliptic ones, are infinitesimal generators of strongly continuous one-parameter semigroups of bounded linear operators on L2(Rn) by first proving Gårding’s Inequality for strongly (ρ,Λ) – elliptic pseudo-differential operators on Rn.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.017 | 0.008 |
Machine scores (provisional)
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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
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