Harmonic functions with traces in Q type spaces related to weights

In this article, via a family of convolution operators {ϕ _t}_t>0 , we characterize the extensions of a class of Q type spaces Q^p,q_K,λ(ℝ^n) related with weights K(· ) . Unlike the classical Q type spaces which are related with power functions, a general weight function K(· ) is short of homogeneity of the dilation, and is not variable-separable. Under several assumptions on the integrability of K(· ) , we establish a Carleson type characterization of Q^p,q_K,λ(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 ℝ^2_+ . For the case n≥ 2 , the family {ϕ _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,λ(ℝ^n) is also obtained, including convolution singular integral operators and fractional integral operators.