<p>We introduce a class of (possibly) degenerate dispersive equations with a real drift. We prove that, under the Hörmander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{\mathcal {T}(t)\}_{t\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mi mathvariant="script">T</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2(\mathbb R^m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>m</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, we prove that for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> the operator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {T}(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">T</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfies a sharp form of dispersive estimate in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>, for any <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1\le p\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and an uncertainty principle.</p>

Schrödinger semigroups and the Hörmander hypoellipticity condition

期刊论文
发表日期：

2025年10月21日

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Nicola Garofalo,
Alessandra Lunardi

摘要

We introduce a class of (possibly) degenerate dispersive equations with a real drift. We prove that, under the Hörmander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup $\{\mathcal {T}(t)\}_{t\ge 0}$ ${T (t)}_{t \geq 0}$ in $L^2(\mathbb R^m)$ $L^{2} (R^{m})$ . Finally, we prove that for $t>0$ $t > 0$ the operator $\mathcal {T}(t)$ $T (t)$ satisfies a sharp form of dispersive estimate in $L^p$ $L^{p}$ , for any $1\le p\le 2$ $1 \leq p \leq 2$ , and an uncertainty principle.