Volume 36, Issue 3
Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

Jiaxi Jiu & Zhongkai Li

Anal. Theory Appl., 36 (2020), pp. 326-347.

Published online: 2020-09

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  • Abstract

We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator $D$ in the upper half-plane $R^{2}_+=R\times(0,\infty)$, where
                           $$(Df)(x)=f'(x)+(\lambda/x)[f(x)-f(-x)]$$
for given $\lambda\ge0$. A $C^2$ function $u$ in $R^{2}_+$ is said to be $\lambda$-harmonic if $(D_x^2+\partial_{y}^2)u=0$. For a $\lambda$-harmonic function $u$ in $R^{2}_+$ and for a subset $E$ of $\partial R^{2}_+=R$ symmetric about $y$-axis, we prove that the following three assertions are equivalent: (i) $u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)<\infty$ for a.e. $x\in E$, where $S$ is a Lusin-type area integral associated with the Dunkl operator $D$.

  • Keywords

Dunkl operator, Dunkl transform, harmonic function, non-tangential limit, area integral.

  • AMS Subject Headings

42B20, 42B25, 42A38, 35G10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-36-326, author = {Jiaxi Jiu , and Zhongkai Li , }, title = {Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {3}, pages = {326--347}, abstract = {

We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator $D$ in the upper half-plane $R^{2}_+=R\times(0,\infty)$, where
                           $$(Df)(x)=f'(x)+(\lambda/x)[f(x)-f(-x)]$$
for given $\lambda\ge0$. A $C^2$ function $u$ in $R^{2}_+$ is said to be $\lambda$-harmonic if $(D_x^2+\partial_{y}^2)u=0$. For a $\lambda$-harmonic function $u$ in $R^{2}_+$ and for a subset $E$ of $\partial R^{2}_+=R$ symmetric about $y$-axis, we prove that the following three assertions are equivalent: (i) $u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)<\infty$ for a.e. $x\in E$, where $S$ is a Lusin-type area integral associated with the Dunkl operator $D$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-SU11}, url = {http://global-sci.org/intro/article_detail/ata/18289.html} }
TY - JOUR T1 - Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator AU - Jiaxi Jiu , AU - Zhongkai Li , JO - Analysis in Theory and Applications VL - 3 SP - 326 EP - 347 PY - 2020 DA - 2020/09 SN - 36 DO - http://doi.org/10.4208/ata.OA-SU11 UR - https://global-sci.org/intro/article_detail/ata/18289.html KW - Dunkl operator, Dunkl transform, harmonic function, non-tangential limit, area integral. AB -

We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator $D$ in the upper half-plane $R^{2}_+=R\times(0,\infty)$, where
                           $$(Df)(x)=f'(x)+(\lambda/x)[f(x)-f(-x)]$$
for given $\lambda\ge0$. A $C^2$ function $u$ in $R^{2}_+$ is said to be $\lambda$-harmonic if $(D_x^2+\partial_{y}^2)u=0$. For a $\lambda$-harmonic function $u$ in $R^{2}_+$ and for a subset $E$ of $\partial R^{2}_+=R$ symmetric about $y$-axis, we prove that the following three assertions are equivalent: (i) $u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)<\infty$ for a.e. $x\in E$, where $S$ is a Lusin-type area integral associated with the Dunkl operator $D$.

Jiaxi Jiu & Zhongkai Li. (2020). Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator. Analysis in Theory and Applications. 36 (3). 326-347. doi:10.4208/ata.OA-SU11
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