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