$L^p$ boundedness of discrete singular Radon transforms
$L^p$ boundedness of discrete singular Radon transforms
We prove that if $K:\mathbb {R}^{d_1}\to \mathbb {C}$ is a CalderónâZygmund kernel and $P:\mathbb {R}^{d_1}\to \mathbb {R}^{d_2}$ is a polynomial of degree $A\geq 1$ with real coefficients, then the discrete singular Radon transform operator \begin{equation*} T(f)(x)=\sum _{n\in \mathbb {Z}^{d_1}\setminus \{0\}}f(x-P(n))K(n) \end{equation*} extends to a bounded operator on $L^p(\mathbb {R}^{d_2})$, $1<p<\infty$. …