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Pointwise monotonicity of heat kernels
Diego Alonso-Orán1, Fernando Chamizo2, Ángel D Martínez2
1Institute für Angewandte Mathematik, Universitat Bonn, Endenicher Allee 60, Bonn, 53115 Germany.
Revista Matematica Complutense
|January 24, 2023
Summary
This study presents a simple proof for the radial monotonicity of heat kernels across Euclidean, spherical, and hyperbolic spaces. This finding offers new inequalities for special functions.
Area of Science:
- Differential Geometry
- Analysis
- Mathematical Physics
Background:
- The Cheeger-Yau theorem (1981) established radial monotonicity of heat kernels.
- This property has been recently rediscovered in specific geometric contexts.
- Understanding heat kernel behavior is crucial in various mathematical fields.
Purpose of the Study:
- To provide an elementary proof of the pointwise radial monotonicity property of heat kernels.
- To generalize this property to Euclidean spaces, spheres, and hyperbolic spaces.
- To derive new inequalities involving special functions as a consequence.
Main Methods:
- An elementary, yet non-straightforward, application of the parabolic maximum principle.
- Focus on the monotonicity of the heat kernel originating from special points on revolution hypersurfaces.
Main Results:
- A unified and elementary proof for the radial monotonicity of heat kernels.
- Demonstration that this property holds across diverse geometric spaces.
- Derivation of novel inequalities related to classical special functions.
Conclusions:
- The parabolic maximum principle offers an accessible method for proving heat kernel monotonicity.
- The established monotonicity property has broad applicability in geometric analysis.
- The derived inequalities provide new tools for studying special functions.
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