Harmonic extension of Q-type space related to logarithmic functions

Abstract

This paper studies a class of Q-type spaces Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)related to logarithmic functions. We first investigate some basic properties of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). Further, by the aid of Poisson integral and harmonic function spaces H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right), the harmonic extension of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)and the boundary value problem of H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right)are obtained.

Keywords

Q type space / Poisson integral / harmonic extension

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