Harmonic functions with traces in Q type spaces related to weights

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