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