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Singular integral operators associated to curves with rational components

Authors:	Magali Folch-Gabayet James Wright

Institution:	Instituto de Matemáticas, UNAM, Area de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, México, D.F. 04510 ; School of Mathematics, University of Edinburgh, JCMB, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland

Abstract:	We prove $L^p ({\mathbb{R}}^n), 1<p<\infty$ , bounds for $\displaystyle Hf(x) = p.v. \int_{-\infty}^{\infty} f(x_1 - R_1 (t), \ldots , x_n - R_n (t) ) \, dt/t$ and $\displaystyle Mf(x) = \sup_{h>0} {1\over h} \int_{0}^{h} \vert f(x_1 - R_1 (t), \ldots , x_n - R_n (t) )\vert \, dt$ where $R_j (t) = P_j(t)/Q_j(t), j=1,2,\ldots, n$ , are rational functions. Our bounds depend only on the degrees of the polynomials and, in particular, they do not depend on the coefficients of these polynomials.

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