Extension of Q type spaces related with weights via s-harmonic equations and applications

Abstract

In this paper, we use the s-harmonic functions which are the solutions to the following equations{div(t1−s∇u)=0,(x,t)∈R+n+1;u(x,0)=f(x),x∈Rn to characterize a new class of Q-type spaces QK,λp(Rn) which are related to weight functions K. By the aid of the fractional Poisson kernel pts(⋅), s∈(0,2), we establish a Carleson type extension of QK,λp(Rn) to the space HKp,λ(R+n+1). As applications, the extension results can be applied to the Q type spaces related with logarithmic functions. Moreover, the boundedness of convolution singular integrals on QK,λp(Rn) and the s-harmonic extensions of Campanato-Sobolev classes are also considered, respectively.