An Extension of Riesz Transform

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