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Optimal in tail Hölder estimates for weak solutions of the nonlocal parabolic p-Laplace equations on the Heisenberg group

Debraj Kar

TL;DR

This work establishes local Hölder continuity for weak solutions to the nonlocal parabolic $p$-Laplacian on the Heisenberg group under an optimal tail condition. By deriving a time-dependent Caccioppoli inequality and developing De Giorgi-type lemmas, forward-in-time estimates, and shrinking lemmas, the authors handle both singular ($1<p\le 2$) and degenerate ($p>2$) regimes via intrinsic scaling and tail control. The analysis hinges on a carefully controlled nonlocal tail, fractional Sobolev framework, and geometric properties of the Heisenberg group, culminating in explicit oscillation decay bounds on intrinsic cylinders. The results extend prior Euclidean and Heisenberg-group regularity theories by proving Hölder regularity under the optimal tail condition in the nonlocal parabolic setting, with potential implications for jump-diffusion models in sub-Riemannian spaces.

Abstract

We prove the Hölder continuity for weak solutions to parabolic p-Laplace equations on the Heisenberg group. We deduce this result while considering an optimal tail condition.

Optimal in tail Hölder estimates for weak solutions of the nonlocal parabolic p-Laplace equations on the Heisenberg group

TL;DR

This work establishes local Hölder continuity for weak solutions to the nonlocal parabolic -Laplacian on the Heisenberg group under an optimal tail condition. By deriving a time-dependent Caccioppoli inequality and developing De Giorgi-type lemmas, forward-in-time estimates, and shrinking lemmas, the authors handle both singular () and degenerate () regimes via intrinsic scaling and tail control. The analysis hinges on a carefully controlled nonlocal tail, fractional Sobolev framework, and geometric properties of the Heisenberg group, culminating in explicit oscillation decay bounds on intrinsic cylinders. The results extend prior Euclidean and Heisenberg-group regularity theories by proving Hölder regularity under the optimal tail condition in the nonlocal parabolic setting, with potential implications for jump-diffusion models in sub-Riemannian spaces.

Abstract

We prove the Hölder continuity for weak solutions to parabolic p-Laplace equations on the Heisenberg group. We deduce this result while considering an optimal tail condition.
Paper Structure (28 sections, 11 theorems, 204 equations)

This paper contains 28 sections, 11 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Notations, Definitions and Main Result
  3. Notations:
  4. Definitions:
  5. Main Result :
  6. Background and Novelty
  7. Notes on Heisenberg Group
  8. Auxiliary lemmas
  9. Plan of the Paper
  10. Acknowledgments
  11. Caccioppoli inequality
  12. Preliminary Results
  13. De Giorgi Lemma
  14. De Giorgi Lemma : Forward in Time
  15. Measure Theoretical Information Forward in Time
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $p>1,T>0$ and $s\in(0,1)$. Let $u$ be locally bounded weak solution to (main) in $\Omega_T$, where $\Omega$ is a bounded open subset of $\mathbb{H}^N$. Moreover, assume that for some $\varepsilon>0$, Then $u$ is locally Hölder continuous in $\Omega_T$. More precisely, there exists some constants $\gamma,\tilde{\gamma},\lambda$ and $\eta\in (0,1)$ depending upon data, such that for any $0<r<R<

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 1.3
  • Remark 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Proposition 1.7
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • ...and 9 more