Universal L-p improving for averages along polynomial curves in low dimensions

We prove sharp L-p -> L-q estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree. Our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to M. Christ. Almost sharp Lorentz space estimates are obtained as well. (C) 2009 Elsevier Inc. All rights reserved.