Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pointwise convergence for the heat equation on tori $\mathbb T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$

Divyang G. Bhimani, Rupak K. Dalai

TL;DR

This work determines exactly which weighted Lebesgue spaces $L^p_v$ on the torus $\mathbb{T}^n$ and on the waveguide $\mathbb{T}^n\times\mathbb{R}^m$ ensure that the heat equation solution converges pointwise to the initial data as $t\to 0^+$. The authors connect pointwise convergence to sharp boundedness properties of the associated heat-kernel maximal operators, and introduce weight classes $D^{T}_p$ and $D^{WG}_p$ that precisely characterize these bounds. The main contributions include a complete characterization of these weight classes, a proof of pointwise a.e. convergence for the torus and waveguide cases, and the first results establishing analogous convergence criteria on the waveguide manifold. The results have implications for weighted harmonic analysis on compact and semi-periodic spaces and extend the understanding of maximal-operator methods in non-Euclidean settings.

Abstract

We completely characterize the weighted Lebesgue spaces on the torus $\mathbb T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$ for which the solutions of the heat equation converge pointwise (as time tends to zero) to the initial data. In the process, we also characterize the weighted Lebesgue spaces for the boundedness of maximal operators on the torus and waveguide manifold, which may be of independent interest.

Pointwise convergence for the heat equation on tori $\mathbb T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$

TL;DR

This work determines exactly which weighted Lebesgue spaces on the torus and on the waveguide ensure that the heat equation solution converges pointwise to the initial data as . The authors connect pointwise convergence to sharp boundedness properties of the associated heat-kernel maximal operators, and introduce weight classes and that precisely characterize these bounds. The main contributions include a complete characterization of these weight classes, a proof of pointwise a.e. convergence for the torus and waveguide cases, and the first results establishing analogous convergence criteria on the waveguide manifold. The results have implications for weighted harmonic analysis on compact and semi-periodic spaces and extend the understanding of maximal-operator methods in non-Euclidean settings.

Abstract

We completely characterize the weighted Lebesgue spaces on the torus and waveguide manifold for which the solutions of the heat equation converge pointwise (as time tends to zero) to the initial data. In the process, we also characterize the weighted Lebesgue spaces for the boundedness of maximal operators on the torus and waveguide manifold, which may be of independent interest.
Paper Structure (11 sections, 12 theorems, 145 equations, 1 figure)

This paper contains 11 sections, 12 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Heat equation on tori $\mathbb T^n$
  3. Heat equation on waveguide manifold $\mathbb T^n \times \mathbb R^m$
  4. Comments on Theorems \ref{['th1']} and \ref{['th-r1']}
  5. Preliminaries
  6. Heat kernel estimates
  7. Maximal operator estimates on $\mathbb T^n$ and $\mathbb T^n \times \mathbb R^m$
  8. Proof of Main Results
  9. Characterization of weight classes $D^{T}_p$ and $D^{WG}_p$
  10. Proof of Theorem \ref{['th1']}
  11. Proof of Theorem \ref{['th-r1']}

Key Result

Theorem 1.2

Let $v$ be a strictly positive weight on the torus $\mathbb T^n$ and $f\in L^p_v(\mathbb T^n).$ Suppose $0<T< \frac{1}{2}.$ Then $v\in D^{T}_p$ if and only if $u(x, t)$ is a solution of (ht) for $t\in (0,T],$ and

Figures (1)

  • Figure 1: We have drawn the graphs of $A_k$ and $B_k$ in the case of $\mathbb T\times\mathbb R$. On the left side, we have shown the disjoint annuli $A_k$, while on the right side, we have illustrated the balls $B_k$ containing $A_k$.

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1: see Theorem 1.1 in MR1240931
  • Lemma 2.2: Kolmogorov inequality, see Theorem 3.3.1, p. 59 in MR0596037
  • ...and 24 more