Sobolev improving for averages over curves in R4

Abstract We study L p -Sobolev improving for averaging operators A γ given by convolution with a compactly supported smooth density μ γ on a non-degenerate curve. In particular, in 4 dimensions we show that A γ maps L p ( R 4 ) to the Sobolev space L 1 / p p ( R 4 ) for all 6 p ∞ . This implies the complete optimal range of L p -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 μ γ . In higher dimensions, a new non-trivial necessary condition for L p ( R n ) → L 1 / p p ( R n ) boundedness is obtained, which motivates a conjectural range of estimates.