{"title":"Green函数高可积性的Fabes-Stroock方法及Ld漂移下的ABP估计","authors":"Pilgyu Jung , Kwan Woo","doi":"10.1016/j.matpur.2025.103805","DOIUrl":null,"url":null,"abstract":"<div><div>We explore the higher integrability of Green's functions associated with the second-order elliptic equation <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>f</mi></math></span> in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term <span><math><mi>b</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> and the source term <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for some <span><math><mi>p</mi><mo><</mo><mi>d</mi></math></span>. This provides an alternative and analytic proof of a result by N.V. Krylov (<em>Ann. Probab.</em>, 2021) concerning <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (<em>Duke Math. J.</em>, 1984).</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103805"},"PeriodicalIF":2.3000,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with Ld drift\",\"authors\":\"Pilgyu Jung , Kwan Woo\",\"doi\":\"10.1016/j.matpur.2025.103805\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We explore the higher integrability of Green's functions associated with the second-order elliptic equation <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>f</mi></math></span> in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term <span><math><mi>b</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> and the source term <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for some <span><math><mi>p</mi><mo><</mo><mi>d</mi></math></span>. This provides an alternative and analytic proof of a result by N.V. Krylov (<em>Ann. Probab.</em>, 2021) concerning <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (<em>Duke Math. J.</em>, 1984).</div></div>\",\"PeriodicalId\":51071,\"journal\":{\"name\":\"Journal de Mathematiques Pures et Appliquees\",\"volume\":\"204 \",\"pages\":\"Article 103805\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal de Mathematiques Pures et Appliquees\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021782425001497\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782425001497","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with Ld drift
We explore the higher integrability of Green's functions associated with the second-order elliptic equation in a bounded domain , and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term in and the source term for some . This provides an alternative and analytic proof of a result by N.V. Krylov (Ann. Probab., 2021) concerning drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (Duke Math. J., 1984).
期刊介绍：
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.
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