| Summary: | 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.
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