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A Volterra Calculus for Lie Groupoids

Karsten Bohlen

TL;DR

This work develops a Volterra calculus on Lie groupoids to address parabolic PDEs, enabling a rigorous analysis of heat kernels on singular spaces. By introducing anisotropic symbol classes $S_{ai,m}(\mathcal{A}^* \oplus \mathbb{R}_M)$ and a quantization $\mathrm{Op}_{ai}$, the authors construct a closed, composable calculus with Volterra properties and establish a parametrix theory via invertible Volterra principal symbols. A parabolic adiabatic deformation groupoid is built to realize 1-parameter families $T_{\hbar}$ and to study short-time heat kernel asymptotics, yielding a diagonal expansion $k_{t|\Delta} \sim \sum_{j\ge0} t^{(j-d)/m} q_j$; the analysis further shows that logarithmic terms can be controlled, leading to a clean expansion with coefficients depending on the base point. The results provide a robust framework for analyzing heat flows and fundamental solutions on manifolds with corners and other singular geometries, with potential applications to longitudinal heat flows and index theory in the groupoid setting.

Abstract

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

A Volterra Calculus for Lie Groupoids

TL;DR

This work develops a Volterra calculus on Lie groupoids to address parabolic PDEs, enabling a rigorous analysis of heat kernels on singular spaces. By introducing anisotropic symbol classes and a quantization , the authors construct a closed, composable calculus with Volterra properties and establish a parametrix theory via invertible Volterra principal symbols. A parabolic adiabatic deformation groupoid is built to realize 1-parameter families and to study short-time heat kernel asymptotics, yielding a diagonal expansion ; the analysis further shows that logarithmic terms can be controlled, leading to a clean expansion with coefficients depending on the base point. The results provide a robust framework for analyzing heat flows and fundamental solutions on manifolds with corners and other singular geometries, with potential applications to longitudinal heat flows and index theory in the groupoid setting.

Abstract

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the -manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate inner product.

Paper Structure

This paper contains 6 sections, 5 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. The heat semigroup
  3. Anisotropic Operators
  4. Short time asymptotic expansion
  5. Parabolic adiabatic deformation groupoid
  6. Standard coordinates and lift

Key Result

Proposition 2.2

Let $Q \in \Psi_{\mathcal{G},c}^m(r^*E)$ with $m > 0$ be symmetric on $C_c^\infty(\mathcal{G} ,r^*E)$, $\mathcal{G}$-elliptic, and nonnegative. Then its closure on $H^0$ is regular selfadjoint with $\mathrm{spec}(Q) \subset [0,\infty)$. For a given $k \in \mathbb{N}$ and any $u_0 \in H^k$, the initi has a unique solution in $\bigcap_{0 \leq j \leq k} C^j([0, \infty), H^{k-j})$, given by $u(t) = \e

Theorems & Definitions (14)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Lemma 3.2: cf. Greiner
  • proof : Proof sketch
  • Definition 3.3
  • Proposition 3.4
  • Definition 3.5
  • Remark 3.6
  • ...and 4 more