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Some remarks on boundary operators of Bessel extensions

The first author is supported in part by the Marsden Fund Council from New Zealand Government funding, managed by the Royal Society of New Zealand. The second author is supported in part by the Taiwan Ministry of Science and Technology under research grants 103-2115-M-009-016-MY2 and 105-2115-M-009-004-MY2.
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  • 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 $s=k ∈ \mathbb{N}$ .

    Mathematics Subject Classification: Primary: 35J70; Secondary: 47D03, 33C10.


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