Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators
Let be the high-order Schrödinger operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo>&l...
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MDPI AG
2022-07-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/15/2600 |
| _version_ | 1797413147016953856 |
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| author | Wei Chen Chao Zhang |
| author_facet | Wei Chen Chao Zhang |
| author_sort | Wei Chen |
| collection | DOAJ |
| description | Let be the high-order Schrödinger operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>V</mi><mn>2</mn></msup><mo>,</mo></mrow></semantics></math></inline-formula> where <i>V</i> is a non-negative potential satisfying the reverse Hölder inequality (<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></semantics></math></inline-formula>), with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mi>n</mi><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></semantics></math></inline-formula>. In this paper, we prove that when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>2</mn><mo>−</mo><mi>n</mi><mo>/</mo><mi>q</mi></mrow></semantics></math></inline-formula>, the adapted Lipschitz spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mo>Λ</mo><mrow><mi>α</mi><mo>/</mo><mn>4</mn></mrow><mi mathvariant="script">L</mi></msubsup></semantics></math></inline-formula> we considered are equivalent to the Lipschitz space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>C</mi><mrow><mi>L</mi></mrow><mi>α</mi></msubsup></semantics></math></inline-formula> associated to the Schrödinger operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>=</mo><mo>−</mo><mo>Δ</mo><mo>+</mo><mi>V</mi></mrow></semantics></math></inline-formula>. In order to obtain this characterization, we should make use of some of the results associated to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula>. We also prove the regularity properties of fractional powers (positive and negative) of the operator ℒ, Schrödinger Riesz transforms, Bessel potentials and multipliers of the Laplace transforms type associated to the high-order Schrödinger operators. |
| first_indexed | 2024-03-09T05:13:25Z |
| format | Article |
| id | doaj.art-c3d1e18076534dba9a77d2aa48c7e9fc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T05:13:25Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c3d1e18076534dba9a77d2aa48c7e9fc2023-12-03T12:47:23ZengMDPI AGMathematics2227-73902022-07-011015260010.3390/math10152600Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger OperatorsWei Chen0Chao Zhang1School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, ChinaSchool of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, ChinaLet be the high-order Schrödinger operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>V</mi><mn>2</mn></msup><mo>,</mo></mrow></semantics></math></inline-formula> where <i>V</i> is a non-negative potential satisfying the reverse Hölder inequality (<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></semantics></math></inline-formula>), with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mi>n</mi><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></semantics></math></inline-formula>. In this paper, we prove that when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>2</mn><mo>−</mo><mi>n</mi><mo>/</mo><mi>q</mi></mrow></semantics></math></inline-formula>, the adapted Lipschitz spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mo>Λ</mo><mrow><mi>α</mi><mo>/</mo><mn>4</mn></mrow><mi mathvariant="script">L</mi></msubsup></semantics></math></inline-formula> we considered are equivalent to the Lipschitz space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>C</mi><mrow><mi>L</mi></mrow><mi>α</mi></msubsup></semantics></math></inline-formula> associated to the Schrödinger operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>=</mo><mo>−</mo><mo>Δ</mo><mo>+</mo><mi>V</mi></mrow></semantics></math></inline-formula>. In order to obtain this characterization, we should make use of some of the results associated to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula>. We also prove the regularity properties of fractional powers (positive and negative) of the operator ℒ, Schrödinger Riesz transforms, Bessel potentials and multipliers of the Laplace transforms type associated to the high-order Schrödinger operators.https://www.mdpi.com/2227-7390/10/15/2600heat semigrouphigh-order Schrödinger operatorsLipschitz Hölder Zygmund spacesHölder estimates |
| spellingShingle | Wei Chen Chao Zhang Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators Mathematics heat semigroup high-order Schrödinger operators Lipschitz Hölder Zygmund spaces Hölder estimates |
| title | Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators |
| title_full | Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators |
| title_fullStr | Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators |
| title_full_unstemmed | Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators |
| title_short | Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators |
| title_sort | regularity properties and lipschitz spaces adapted to high order schrodinger operators |
| topic | heat semigroup high-order Schrödinger operators Lipschitz Hölder Zygmund spaces Hölder estimates |
| url | https://www.mdpi.com/2227-7390/10/15/2600 |
| work_keys_str_mv | AT weichen regularitypropertiesandlipschitzspacesadaptedtohighorderschrodingeroperators AT chaozhang regularitypropertiesandlipschitzspacesadaptedtohighorderschrodingeroperators |