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 K , λ p , q (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 K , λ p , q (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 K , λ p , q (R n) is also obtained, including convolution singular integral operators and fractional integral operators. [ABSTRACT FROM AUTHOR]