Some remarks on boundary operators of Bessel extensions

TL;DR

This paper investigates boundary operators related to Bessel-type extensions, revealing how logarithmic scaling captures the loss of analyticity at specific parameter limits.

Contribution

It introduces a logarithmic scaling approach to analyze boundary operators of Bessel extensions, especially at critical parameter values where analyticity fails.

Findings

Logarithmic scaling captures failure of analyticity at s=k in natural numbers.

Boundary operators of Bessel extensions are characterized under this scaling.

The approach provides insights into the structure of these extensions at limiting cases.

Abstract

In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is \[\Delta_x u(x,y) +\frac{1-2s}{y} \frac{\partial u}{\partial y}(x,y)+\frac{\partial^2 u}{\partial y^2}(x,y)=0 \text{ for }x\in\mathbb{R}^d, y>0, \\ u(x,0)=f(x) \text{ for }x\in\mathbb{R}^d. \] In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases $s=k \in \mathbb{N}$.

Full text

See pages 1-17 of Goodman-Spector-Remarks-On-An-Extension-Problem-arxiv.pdf