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

Abstract

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.

21 pages, with revised introduction and updated references

References in corpus (2)

• Quasiextremals for a Radon-like transform
• Restriction of Fourier transforms to curves and related oscillatory integrals

Cited by in corpus (5)

• Endpoint bounds for a generalized {R}adon transform
• On the Oberlin affine curvature condition
• Geometric averaging operators and nonconcentration inequalities
• Convolution with measures on flat curves in low dimensions
• Restriction of Fourier transforms to curves: An endpoint estimate with affine arclength measure