Singular integral operators associated to curves with rational components

Abstract

We prove $L^p ({\mathbb R}^n), 1<p<\infty$, bounds for \[ Hf(x) = p.v. \int _{-\infty }^{\infty } f(x_1 - R_1 (t), \ldots , x_n - R_n (t) ) dt/t \] and \[ Mf(x) = \sup _{h>0} {1\over h} \int _{0}^{h} |f(x_1 - R_1 (t), \ldots , x_n - R_n (t) )| dt \] 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 $P_j, Q_j$ and, in particular, they do not depend on the coefficients of these polynomials.

Locations

• Edinburgh Research Explorer (University of Edinburgh) - View - PDF
• Transactions of the American Mathematical Society - View - PDF