In this note, we generalize some of the geometric results obtained by Korobenko, Sawyer, Rios, and Shen, who studied operators of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\nabla \cdot A\nabla $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mi>A</mi> <mi mathvariant="normal">∇</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A(x)\approx \{1,f^2(x_1)\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>f</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> to the 3D case where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${A(x)\approx \textrm{diag}\{1,f^2(x_1), g^2(x_1)\}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≈</mo> <mtext>diag</mtext> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>f</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msup> <mi>g</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. More precisely, we make explicit calculations of the geodesics in the Carnot–Carathéodory space associated to A and provide estimates on the Lebesgue measures of metric balls centered at the origin in that space.

Carnot–Carathéodory metrics associated to degenerate elliptic operators in three dimensions

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发表日期：

2026年2月20日

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Florian Meister,
Olive Ross,
Lyudmila Korobenko

摘要

In this note, we generalize some of the geometric results obtained by Korobenko, Sawyer, Rios, and Shen, who studied operators of the form $\nabla \cdot A\nabla $ $\nabla \cdot A \nabla$ with $A(x)\approx \{1,f^2(x_1)\}$ $A (x) \approx {1, f^{2} (x_{1})}$ to the 3D case where ${A(x)\approx \textrm{diag}\{1,f^2(x_1), g^2(x_1)\}}$ $A (x) \approx diag {1, f^{2} (x_{1}), g^{2} (x_{1})}$ . More precisely, we make explicit calculations of the geodesics in the Carnot–Carathéodory space associated to A and provide estimates on the Lebesgue measures of metric balls centered at the origin in that space.