Harmonic functions with traces in Q type spaces related to weights

In this article, via a family of convolution operators \(\{\phi _t\}_{t>0}\) ${\{{\varphi }_t\}}_{t>0}$ , we characterize the extensions of a class of Q type spaces \(Q^{p,q}_{K,\lambda }(\mathbb {R}^n)\) $Q_{K,\lambda }^{p,q}({\mathbb{R}}^n)$ related with weights \(K(\cdot )\) $K(·)$ . Unlike the classical Q type spaces which are related with power functions, a general weight function \(K(\cdot )\) $K(·)$ is short of homogeneity of the dilation, and is not variable-separable. Under several assumptions on the integrability of \(K(\cdot )\) $K(·)$ , we establish a Carleson type characterization of \(Q^{p,q}_{K,\lambda }(\mathbb {R}^n)\) $Q_{K,\lambda }^{p,q}({\mathbb{R}}^n)$ . We provide several applications. For the spatial dimension \(n=1\) $n=1$ , such an extension result indicates a boundary characterization of a class of analytic functions on \(\mathbb R^{2}_{+}\) ${\mathbb{R}}_{+}^2$ . For the case \(n\ge 2\) $n\ge 2$ , the family \(\{\phi _t\}_{t>0}\) ${\{{\varphi }_t\}}_{t>0}$ can be seen as a generalization of the fundamental solutions to fractional heat equations, Caffarelli–Silvestre extensions and time-space fractional equations, respectively. Moreover, the boundedness of convolution operators on \(Q^{p,q}_{K,\lambda }(\mathbb {R}^n)\) $Q_{K,\lambda }^{p,q}({\mathbb{R}}^n)$ is also obtained, including convolution singular integral operators and fractional integral operators.