In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schrödinger equation on a Riemannian manifold M with nonnegative Ricci curvature, where the potential term V is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^4$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> on M. We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> on M. In the particular case that V is the quadratic potential <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V(x)=|x|^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and M is the Euclidean space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schrödinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Delta - x\cdot \nabla $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mo>-</mo> <mi>x</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> </mrow> </math></EquationSource> </InlineEquation> with quadratic potential in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>.

A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators

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2026年2月28日

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Ben Andrews,
Daniel Hauer,
Jessica Slegers

摘要

In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schrödinger equation on a Riemannian manifold M with nonnegative Ricci curvature, where the potential term V is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least $C^4$ $C^{4}$ on M. We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only $C^2$ $C^{2}$ on M. In the particular case that V is the quadratic potential $V(x)=|x|^2$ $V (x) = {| x |}^{2}$ and M is the Euclidean space $\mathbb {R}^d$ $R^{d}$ , we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schrödinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator $\Delta - x\cdot \nabla $ $Δ - x \cdot \nabla$ with quadratic potential in $\mathbb {R}^d$ $R^{d}$ .