# 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.

## Key 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 $s=k \in \mathbb{N}$.