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An extension problem of higher order operators and operators of logarithmic type via renormalization

David Lee

TL;DR

This work extends the Caffarelli–Silvestre extension framework to higher-order and logarithmic-type operators using renormalization. The main achievement is a precise classification of the renormalized Neumann-to-Dirichlet map: for non-integer $\nu$, it equals a constant multiple of $(-\Delta)^\nu$, while for integer $\nu$ a logarithmic correction involving the digamma function appears, reflecting a renormalization anomaly. The analysis connects to the modulated energy in Coulomb gases via Hadamard regularization and provides two renormalization-based perspectives on the logarithmic Laplacian extension problem. The results yield a unified viewpoint for higher-order and zero-order (logarithmic) extension problems and open avenues for further extensions to other operators via renormalized boundary operators.

Abstract

We introduce a method of obtaining a higher order extension problem, á la Caffarelli-Silvestre, utilizing ideas from renormalization. Moreover, we give an alternative perspective of the recently developed extension problem for the logarithmic laplacian developed by Chen, Hauer and Weth (2023) [arXiv:2312.15689].

An extension problem of higher order operators and operators of logarithmic type via renormalization

TL;DR

This work extends the Caffarelli–Silvestre extension framework to higher-order and logarithmic-type operators using renormalization. The main achievement is a precise classification of the renormalized Neumann-to-Dirichlet map: for non-integer , it equals a constant multiple of , while for integer a logarithmic correction involving the digamma function appears, reflecting a renormalization anomaly. The analysis connects to the modulated energy in Coulomb gases via Hadamard regularization and provides two renormalization-based perspectives on the logarithmic Laplacian extension problem. The results yield a unified viewpoint for higher-order and zero-order (logarithmic) extension problems and open avenues for further extensions to other operators via renormalized boundary operators.

Abstract

We introduce a method of obtaining a higher order extension problem, á la Caffarelli-Silvestre, utilizing ideas from renormalization. Moreover, we give an alternative perspective of the recently developed extension problem for the logarithmic laplacian developed by Chen, Hauer and Weth (2023) [arXiv:2312.15689].

Paper Structure

This paper contains 12 sections, 6 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Main Result
  4. Outline of the article
  5. Some context behind our result
  6. The modulated energy for Coulomb gases
  7. Alternative perspectives on the logarithmic extension problem of Chen, Hauer and Weth
  8. A quick review on the modified Bessel function of first and second kind
  9. Proof of the main result
  10. Solving the Neumann problem
  11. Proof of Proposition \ref{['prop:renormalized_lim']}
  12. Proof of Proposition \ref{['prop:renormalized_lim_1']}

Key Result

Theorem 1.5

We have the following classification of the renormalized Neumann-to-Dirichlet operator $\Lambda^{\mathcal{R}}_{\nu}:\mathcal{S}(\mathbb{R}^d)\rightarrow \mathcal{S}'(\mathbb{R}^d)$ given by:

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.2
  • Proposition 3.1
  • proof
  • ...and 9 more