Some remarks on boundary operators of Bessel extensions

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