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

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

2008-05-28

arxiv.org/abs/0805.4344v3

Mathematics

Classical Analysis and ODEs

21 pages, with revised introduction and updated references

Scientific paper

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.

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