Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces
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Abstract
In this paper, we first introduce some new classes of weighted amalgam spaces. Then, we give the weighted strong-type and weak-type estimates for fractional integral operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math> on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mfenced open="[" close="]"><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math> generated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>b</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math> are established as well. In addition, we are going to study related problems about two-weight, weak-type inequalities for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mfenced open="[" close="]"><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math> on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msub></mml:math> and generalized fractional integrals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:msup><mml:mrow><mml:mi>ℒ</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>γ</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> in the context of weighted amalgam spaces, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>γ</mml:mi><mml:mo><</mml:mo><mml:mi>n</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi mathvariant="script">L</mml:mi></mml:math> is the infinitesimal generator of an analytic semigroup on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mfenced open="(" close=")"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:math> with Gaussian kernel bounds.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 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) | 0.001 | 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