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Endpoint bounds for a generalized Radon transform. (English) Zbl 1174.42010
The author proves that convolution with affine arclength measure on the curve parametrized by $h\left(t\right):=\left(t,{t}^{2},\cdots ,{t}^{n}\right)$ is a bounded operator from ${L}^{p}\left({ℝ}^{n}\right)$ to ${L}^{q}\left({ℝ}^{n}\right)$ for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds. A recent result of S. Dendrinos, N. Laghi and J. Wright [“Universal ${L}^{p}$ improving for averages along polynomial curves in low dimensions, Preprint (2008), arXiv:0805.4344] establishes sharp Lebesgue space bounds(with an accompanying Lorentz space improvement) for convolution with affine arclength measure along polynormial curves in dimensions 2 and 3.

##### MSC:
 42B10 Fourier type transforms, several variables 44A35 Convolution (integral transforms) 44A12 Radon transform