An Extension of Riesz Transform

Abstract

In this paper, we consider the following singular integral \begin{equation*} T_jf(x)=K_j*f(x), K_j(x)=\frac{x_j}{|x|^{n+1-β}}, \end{equation*} where $x\in R^n,0\le {\hat{I}}^2<n,j=1,2,\cdots ,n$. When ${\hat{I}}^2=0$, it corresponds to the Riesz transform. We will make an estimate the $L^q(1<q<\infty )$ norm of $T_{j}f$, which holds uniformly for $0\le {\hat{I}}^2<\frac{n(q-1)}{q}$. In particular, when ${\hat{I}}^2=0$, the strong $(q,q)$ type estimate of the Riesz transform for $1<q<\infty$ is recovered from the obtained estimate.

12 pages