| Summary: | 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>.
|
|---|