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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(t):=(t,t 2 ,,t n ) is a bounded operator from L p ( n ) to L q ( n ) 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.

42B10Fourier type transforms, several variables
44A35Convolution (integral transforms)
44A12Radon transform