Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups

Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>0</mn></msub><mo>,</mo><msub><mi>X</mi><mn>1</mn></m...

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Autor principal: Yuexia Hou
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
nondivergence operator
weighted Sobolev–Morrey estimates
homogeneous group
singular integral
commutators
Mathematics
QA1-939
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spelling oai:doaj.org-article:dd5ddfc7069940909f1db705dffc86a02021-11-25T19:06:25ZWeighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups10.3390/sym131120612073-8994https://doaj.org/article/dd5ddfc7069940909f1db705dffc86a02021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2061https://doaj.org/toc/2073-8994Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>0</mn></msub><mo>,</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>q</mi><mo><</mo><mi>N</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be real vector fields, which are left invariant on homogeneous group <i>G</i>, provided that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mn>0</mn></msub></semantics></math></inline-formula> is homogeneous of degree two and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula> are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>q</mi></munderover></mstyle><mrow><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><msub><mi>X</mi><mi>i</mi></msub><msub><mi>X</mi><mi>j</mi></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><msub><mi>X</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, where the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> satisfies the uniform ellipticity condition on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>q</mi></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.Yuexia HouMDPI AGarticlenondivergence operatorweighted Sobolev–Morrey estimateshomogeneous groupsingular integralcommutatorsMathematicsQA1-939ENSymmetry, Vol 13, Iss 2061, p 2061 (2021)
institution DOAJ
collection DOAJ
language EN
topic nondivergence operator
weighted Sobolev–Morrey estimates
homogeneous group
singular integral
commutators
Mathematics
QA1-939
spellingShingle nondivergence operator
weighted Sobolev–Morrey estimates
homogeneous group
singular integral
commutators
Mathematics
QA1-939
Yuexia Hou
Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups
description Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>0</mn></msub><mo>,</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>q</mi><mo><</mo><mi>N</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be real vector fields, which are left invariant on homogeneous group <i>G</i>, provided that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mn>0</mn></msub></semantics></math></inline-formula> is homogeneous of degree two and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula> are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>q</mi></munderover></mstyle><mrow><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><msub><mi>X</mi><mi>i</mi></msub><msub><mi>X</mi><mi>j</mi></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><msub><mi>X</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, where the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> satisfies the uniform ellipticity condition on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>q</mi></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.
format article
author Yuexia Hou
author_facet Yuexia Hou
author_sort Yuexia Hou
title Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups
title_short Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups
title_full Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups
title_fullStr Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups
title_full_unstemmed Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups
title_sort weighted sobolev–morrey estimates for nondivergence degenerate operators with drift on homogeneous groups
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/dd5ddfc7069940909f1db705dffc86a0
work_keys_str_mv AT yuexiahou weightedsobolevmorreyestimatesfornondivergencedegenerateoperatorswithdriftonhomogeneousgroups
_version_ 1718410310142918656

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