Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces
Assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>μ</mi><mo>)&l...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-03-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/7/1112 |
| _version_ | 1797438544042524672 |
|---|---|
| author | Tianjun Shen Bo Li |
| author_facet | Tianjun Shen Bo Li |
| author_sort | Tianjun Shen |
| collection | DOAJ |
| description | Assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a metric measure space that satisfies a <i>Q</i>-doubling condition with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and supports an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-Poincaré inequality. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>𝓛</mi></semantics></math></inline-formula> be a nonnegative operator generalized by a Dirichlet form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">E</mi></semantics></math></inline-formula> and <i>V</i> be a Muckenhoupt weight belonging to a reverse Hölder class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><msub><mi>H</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mo>(</mo><mi>Q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. In this paper, we consider the Dirichlet problem for the Schrödinger equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msubsup><mo>∂</mo><mi>t</mi><mn>2</mn></msubsup><mi>u</mi><mo>+</mo><mi>𝓛</mi><mi>u</mi><mo>+</mo><mi>V</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> on the upper half-space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>×</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>, which has <i>f</i> as its the boundary value on <i>X</i>. We show that a solution <i>u</i> of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function <i>f</i> such that <i>u</i> can be expressed by the Poisson integral of <i>f</i>. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space <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> to the metric measure space <i>X</i> and improves the reverse Hölder index from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mi>Q</mi></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mo>(</mo><mi>Q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T11:38:28Z |
| format | Article |
| id | doaj.art-af1eb2015cd248b2b556229635092e9d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:38:28Z |
| publishDate | 2022-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-af1eb2015cd248b2b556229635092e9d2023-11-30T23:37:21ZengMDPI AGMathematics2227-73902022-03-01107111210.3390/math10071112Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure SpacesTianjun Shen0Bo Li1Center for Applied Mathematics, Tianjin University, Tianjin 300072, ChinaCollege of Data Science, Jiaxing University, Jiaxing 314001, ChinaAssume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a metric measure space that satisfies a <i>Q</i>-doubling condition with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and supports an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-Poincaré inequality. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>𝓛</mi></semantics></math></inline-formula> be a nonnegative operator generalized by a Dirichlet form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">E</mi></semantics></math></inline-formula> and <i>V</i> be a Muckenhoupt weight belonging to a reverse Hölder class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><msub><mi>H</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mo>(</mo><mi>Q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. In this paper, we consider the Dirichlet problem for the Schrödinger equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msubsup><mo>∂</mo><mi>t</mi><mn>2</mn></msubsup><mi>u</mi><mo>+</mo><mi>𝓛</mi><mi>u</mi><mo>+</mo><mi>V</mi><mi>u</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> on the upper half-space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>×</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>, which has <i>f</i> as its the boundary value on <i>X</i>. We show that a solution <i>u</i> of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function <i>f</i> such that <i>u</i> can be expressed by the Poisson integral of <i>f</i>. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space <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> to the metric measure space <i>X</i> and improves the reverse Hölder index from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mi>Q</mi></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mo>(</mo><mi>Q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/10/7/1112Schrödinger equationMorrey spaceDirichlet problemmetric measure space |
| spellingShingle | Tianjun Shen Bo Li Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces Mathematics Schrödinger equation Morrey space Dirichlet problem metric measure space |
| title | Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces |
| title_full | Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces |
| title_fullStr | Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces |
| title_full_unstemmed | Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces |
| title_short | Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces |
| title_sort | schrodinger harmonic functions with morrey traces on dirichlet metric measure spaces |
| topic | Schrödinger equation Morrey space Dirichlet problem metric measure space |
| url | https://www.mdpi.com/2227-7390/10/7/1112 |
| work_keys_str_mv | AT tianjunshen schrodingerharmonicfunctionswithmorreytracesondirichletmetricmeasurespaces AT boli schrodingerharmonicfunctionswithmorreytracesondirichletmetricmeasurespaces |