Record W3012166630 · doi:10.1155/2020/6907170

Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

2020· article· lv· W3012166630 on OpenAlex

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fundA Canadian funder is recorded on the work.

Bibliographic record

VenueJournal of Function Spaces · 2020

Typearticle

Languagelv

FieldMathematics

TopicAdvanced Harmonic Analysis Research

Canadian institutionsMemorial University of Newfoundland

FundersMemorial University of Newfoundland

KeywordsAlgorithmArtificial intelligenceComputer science

DOI10.1155/2020/6907170

Abstract

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Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>ℒ</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:mrow></mml:math> be a Schrödinger operator on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>d</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>Δ</mml:mi></mml:math> is the Laplacian operator on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>, and the nonnegative potential V belongs to the reverse Hölder class <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:msub><mml:mrow><mml:mtext>RH</mml:mtext></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>s</mml:mi><mml:mo>≥</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math>. For given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>α</mml:mi><mml:mo><</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math>, the fractional integrals associated with the Schrödinger operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi>ℒ</mml:mi></mml:math> is defined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>ℒ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:math>. Suppose that b is a locally integrable function on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mrow><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> and the commutator generated by b and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> is defined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mrow><mml:mfenced open="[" close="]" separators="|"><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>α</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>b</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>⋅</mml:mo><mml:msub><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>b</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:mfenced><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced></mml:mrow></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="M14"><mml:mrow><mml:msub><mml:mrow><mml:mtext>RH</mml:mtext></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mrow><mml:mi>s</mml:mi><mml:mo>≥</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math>. Then, we will establish the boundedness properties of the fractional integrals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M16"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M17"><mml:mrow><mml:mfenced open="[" close="]" separators="|"><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>α</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math> in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M18"><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:mrow><mml:mtext>BMO</mml:mtext><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M20"><mml:mrow><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:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math> corresponding to the classical case (that is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M21"><mml:mrow><mml:mi>V</mml:mi><mml:mo>≡</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>).

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Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.001

metaresearch head score (Gemma)0.002

Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated

Candidate categoriesMeta-epidemiology (narrow), Insufficient payload (model declined to judge)

Consensus categoriesnone

DomainCandidate signal: none · Consensus signal: none

Study designCandidate signal: Not applicable · Consensus signal: none

GenreCandidate signal: Empirical · Consensus signal: Empirical

Teacher disagreement score0.265

Threshold uncertainty score1.000

Codex and Gemma teacher scores by category

Category	Codex	Gemma
Metaresearch	0.001	0.002
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

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.

Opus teacher head0.037

GPT teacher head0.326

Teacher spread0.289 · how far apart the two teachers sit on this one work

Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it

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Published2020

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