An Extension of Riesz Transform

TL;DR

This paper extends the Riesz transform by analyzing a family of singular integrals with a parameter, providing uniform $L^q$ bounds that generalize known results for the classical Riesz transform.

Contribution

It introduces a generalized singular integral operator extending the Riesz transform and establishes uniform $L^q$ bounds across a range of the parameter $eta$.

Findings

Established uniform $L^q$ bounds for the extended Riesz transform.

Recovered classical Riesz transform estimates as a special case.

Provided bounds valid for a range of $eta$ values.

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-\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 $L^q (1<q<\infty)$ norm of $T_jf$, 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 $1<q<\infty$ is recovered from the obtained estimate.