Endpoint bounds for a generalized {R}adon transform

Abstract

We prove that convolution with affine arclength measure on the curve parametrized by $h(t):=(t,t^2,...,t^n)$ is a bounded operator from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds.

24 pages, The final version of this article will appear in the Journal of the London Math. Soc

References in corpus (5)

• Quasiextremals for a Radon-like transform
• $L^p$ improving bounds for averages along curves
• The Fourier extension operator on large spheres and related oscillatory integrals
• Universal L^p improving for averages along polynomial curves in low dimensions
• $L^p$-improving estimates for averages on polynomial curves

Cited by in corpus (7)

• Quasiextremals for a Radon-like transform
• On extremals for a Radon-like transform
• $L^p$-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality
• Sharp smoothing properties of averages over curves
• Convolution with measures on flat curves in low dimensions
• Uniform $L_x^p-L_{x,r}^q$ Improving for Dilated Averages over Polynomial Curves
• $L^p$ improving multilinear Radon-like transforms