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boundedness of discrete singular Radon transforms

Authors:	Alexandru D Ionescu Stephen Wainger

Institution:	Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Van Vleck Hall, Madison, Wisconsin 53706 ; Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Van Vleck Hall, Madison, Wisconsin 53706-1313

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 $\displaystyle T(f)(x)=\sum_{n\in\mathbb{Z}^{d_1}\setminus\{0\}}f(x-P(n))K(n)$ 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.

Keywords:	Singular Radon transforms discrete operators orthogonality square functions exponential sums the circle method

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