Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms
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Abstract
Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi mathvariant="script">L</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:math>be a Schrödinger operator, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:math>is the Laplacian on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>and the nonnegative potential<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:math>belongs to the reverse Hölder class<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math>for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>q</mml:mi><mml:mo>≥</mml:mo><mml:mi>d</mml:mi></mml:math>. The Riesz transform associated with the operator<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi mathvariant="script">L</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:math>is denoted by<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi mathvariant="script">R</mml:mi><mml:mo>=</mml:mo><mml:mo>∇</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo>+</mml:mo><mml:mi>V</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup></mml:math>and the dual Riesz transform is denoted by<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:msup><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo>+</mml:mo><mml:mi>V</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo>∇</mml:mo></mml:math>. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math>for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi>q</mml:mi><mml:mo>≥</mml:mo><mml:mi>d</mml:mi></mml:math>. Then we will establish the mapping properties of the operator<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:math>and its adjoint<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mo stretchy="false">[</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">R</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mo stretchy="false">[</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo></mml:math>are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger than<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M16"><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M17"><mml:mi mathvariant="normal">B</mml:mi><mml:mi mathvariant="normal">M</mml:mi><mml:mi mathvariant="normal">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M18"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>κ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>corresponding to the classical Riesz transforms (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:mi>V</mml:mi><mml:mo>≡</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math>).
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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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 it