𝐿^{𝑝} boundedness of discrete singular Radon transforms

Ionescu, Alexandru; Wainger, Stephen

doi:10.1090/S0894-0347-05-00508-4

$L^p$ boundedness of discrete singular Radon transforms
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by Alexandru D. Ionescu and Stephen Wainger

J. Amer. Math. Soc. 19 (2006), 357-383

DOI: https://doi.org/10.1090/S0894-0347-05-00508-4

Published electronically: October 24, 2005

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Abstract:

We prove that if $K:\mathbb {R}^{d_1}\to \mathbb {C}$ is a Calderón–Zygmund 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 $L^p(\mathbb {R}^{d_2})$, $1<p<\infty$. This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.

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