We study Lp-Sobolev improving for averaging operators Aγ given by convolution with a compactly supported smooth density μγ on a non-degenerate curve. In particular, in 4 dimensions we show that Aγ maps Lp(R4) the Sobolev space Lp1/p(R4) for all 6<p<∞. This implies the complete optimal range of Lp-Sobolev estimates, except possibly for certain endpoint cases. The proof relies on decoupling inequalities for a family of cones which decompose the wave front set of μγ. In higher dimensions, a new non-trivial necessary condition for Lp(Rn)→Lp1/p(Rn) boundedness is obtained, which motivates a conjectural range of estimates.
|Number of pages||60|
|Journal||Advances in Mathematics|
|Publication status||Accepted/In press - 5 Oct 2021|