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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| Format: | Article |
| Language: | English |
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MDPI AG
2021-11-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/11/2061 |
| _version_ | 1797508402526552064 |
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| author | Yuexia Hou |
| author_facet | Yuexia Hou |
| author_sort | Yuexia Hou |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:01:35Z |
| format | Article |
| id | doaj.art-dd5ddfc7069940909f1db705dffc86a0 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T05:01:35Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-dd5ddfc7069940909f1db705dffc86a02023-11-23T01:44:14ZengMDPI AGSymmetry2073-89942021-11-011311206110.3390/sym13112061Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous GroupsYuexia Hou0School of Science, Xi’an Shiyou University, Xi’an 710065, ChinaLet <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.https://www.mdpi.com/2073-8994/13/11/2061nondivergence operatorweighted Sobolev–Morrey estimateshomogeneous groupsingular integralcommutators |
| spellingShingle | Yuexia Hou Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups Symmetry nondivergence operator weighted Sobolev–Morrey estimates homogeneous group singular integral commutators |
| title | 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_short | 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 |
| topic | nondivergence operator weighted Sobolev–Morrey estimates homogeneous group singular integral commutators |
| url | https://www.mdpi.com/2073-8994/13/11/2061 |
| work_keys_str_mv | AT yuexiahou weightedsobolevmorreyestimatesfornondivergencedegenerateoperatorswithdriftonhomogeneousgroups |