Some remarks on boundary operators of Bessel extensions
Jesse Goodman Daniel Spector
In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is
$\begin{align*}Δ_x u(x, y) +\frac{1-2s}{y} \frac{\partial u}{\partial y}(x, y)+\frac{\partial^2 u}{\partial y^2}(x, y)&=0 &&\text{for }x∈\mathbb{R}^d, y>0, \\ u(x, 0)&=f(x) &&\text{for }x∈\mathbb{R}^d.\end{align*}$
In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases
$s=k ∈ \mathbb{N}$
keywords: Boundary operator Littlewood-Paley extension Bessel functions functional calculus Laplacian

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