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A unified framework for pointwise convergence to the initial data of heat equations in metric measure spaces

Divyang G. Bhimani, Anup Biswas, Rupak K. Dalai

TL;DR

This work develops a general framework for guaranteeing almost everywhere pointwise convergence of heat semigroup solutions to their initial data on doubling metric measure spaces. By imposing structural kernel assumptions and introducing the weight class $D_p$ together with a local maximal operator, the authors characterize when $S_t f$ converges to $f$ as $t\to 0$ for $f$ in weighted spaces, and they establish strong boundedness results via vector-valued maximal inequalities. The framework is demonstrated across a wide range of operators, including gradient-perturbed and purely nonlocal generators, mixed local-nonlocal operators, Laplace–Beltrami operators on manifolds, the Dunkl Laplacian, and Hardy potentials, as well as nonhomogeneous and nonlinear problems. They also extend convergence results to nonhomogeneous equations and nonlinear heat equations, providing applications to nonlinear diffusion with power-type nonlinearities. The results offer a unified, technically robust pathway to analyze a.e. convergence in settings that arise in probability, analysis, geometry, and mathematical physics.

Abstract

Given a metric measure space $(\mathcal{X}, d, μ)$ satisfying the volume doubling condition, we consider a semigroup $\{S_t\}$ and the associated heat operator. We propose general conditions on the heat kernel so that the solutions of the associated heat equations attain the initial data pointwise. We demonstrate that these conditions are satisfied by a broad class of operators, including the Laplace operators perturbed by a gradient, fractional Laplacian, mixed local-nonlocal operators, Laplacian on Riemannian manifolds, Dunkl Laplacian and many more. In addition, we consider the Laplace operator in $\mathbb{R}^n$ with the Hardy potential and establish a characterization for the pointwise convergence to the initial data. We also prove similar results for the nonhomogeneous equations and showcase an application for the power-type nonlinearities.

A unified framework for pointwise convergence to the initial data of heat equations in metric measure spaces

TL;DR

This work develops a general framework for guaranteeing almost everywhere pointwise convergence of heat semigroup solutions to their initial data on doubling metric measure spaces. By imposing structural kernel assumptions and introducing the weight class together with a local maximal operator, the authors characterize when converges to as for in weighted spaces, and they establish strong boundedness results via vector-valued maximal inequalities. The framework is demonstrated across a wide range of operators, including gradient-perturbed and purely nonlocal generators, mixed local-nonlocal operators, Laplace–Beltrami operators on manifolds, the Dunkl Laplacian, and Hardy potentials, as well as nonhomogeneous and nonlinear problems. They also extend convergence results to nonhomogeneous equations and nonlinear heat equations, providing applications to nonlinear diffusion with power-type nonlinearities. The results offer a unified, technically robust pathway to analyze a.e. convergence in settings that arise in probability, analysis, geometry, and mathematical physics.

Abstract

Given a metric measure space satisfying the volume doubling condition, we consider a semigroup and the associated heat operator. We propose general conditions on the heat kernel so that the solutions of the associated heat equations attain the initial data pointwise. We demonstrate that these conditions are satisfied by a broad class of operators, including the Laplace operators perturbed by a gradient, fractional Laplacian, mixed local-nonlocal operators, Laplacian on Riemannian manifolds, Dunkl Laplacian and many more. In addition, we consider the Laplace operator in with the Hardy potential and establish a characterization for the pointwise convergence to the initial data. We also prove similar results for the nonhomogeneous equations and showcase an application for the power-type nonlinearities.

Paper Structure

This paper contains 13 sections, 16 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Prior work, motivation and new contribution
  3. Proof techniques and novelties
  4. Preliminaries
  5. Notations and Definitions
  6. Two weight problem for the Hardy-Littlewood maximal operator
  7. Proof of Theorem \ref{['T-1.3']}
  8. Examples
  9. Heat semigroups in Probability and Geometry
  10. The Dunkl operator
  11. Laplacian with Hardy potential $\mathcal{L}_b=-\Delta + b |x|^{-2}$
  12. Abstract nonhomogeneous heat equation
  13. An application to the classical nonlinear heat equation

Key Result

Theorem 1.3

Let $(\mathcal{X}, d, \mu)$ be a metric measure space with the volume doubling property. Let $v$ be a positive weight in $\mathcal{X}$, $\{\varphi_t\}$ be the heat kernel satisfying Assumption Assump-1.2 and $1 \leq p<\infty$. Then the following statements are equivalent:

Theorems & Definitions (41)

  • Definition 1.1: Weight class $D_p$
  • Theorem 1.3
  • Proposition 2.1: Boundedness of $\mathcal{M}^{\mathcal{X}}_R$ in metric measure spaces
  • Remark 2.2
  • Lemma 2.3: Theorem 1.1 in TorreaPM
  • Lemma 2.4: Kolmogorov inequality, Theorem 3.3.1, p. 59 in Guzman
  • proof : Proof of Proposition \ref{['blmx']}
  • proof : Proof of Theorem \ref{['T-1.3']}
  • Example 4.1: gradient perturbation of $\Delta$
  • Remark 4.2
  • ...and 31 more