Some remarks on boundary operators of Bessel extensions

Jesse Goodman; Daniel Spector

doi:10.3934/dcdss.2018027

Article Contents

2018, Volume 11, Issue 3: 493-509. Doi: 10.3934/dcdss.2018027

This issue Previous Article Fractional Laplacians, perimeters and heat semigroups in Carnot groups Next Article Existence and multiplicity results for resonant fractional boundary value problems

Some remarks on boundary operators of Bessel extensions

Jesse Goodman^1, and
Daniel Spector^2,3, ,

1.
Department of Statistics, University of Auckland, Private Bag 92019, Victoria Street West, Auckland 1142, New Zealand
2.
Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan
3.
National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd, Taipei, 106, Taiwan

* Corresponding author: dspector@math.nctu.edu.tw.
* Corresponding author: dspector@math.nctu.edu.tw.

Received: May 2017

Revised: August 2017

Published: June 2018

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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Abstract

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:

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

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