Sharp Sobolev regularity of a restricted X-ray transform

Abstract

We study L p -Sobolev regularity estimate for the restricted X-ray transforms generated by a nondegenerate curve. Making use of the inductive strategy in the recent work by the authors [22], we establish the sharp L pregularity estimates for the restricted X-ray transform in R d+1 , d ≥ 3. This extends the result due to Pramanik and Seeger [27] in R 3 to every dimension. We also obtain L p -Sobolev regularity of the convolution averages over curves with the optimal regularity α = 1/p for p > 2(d − 1).2010 Mathematics Subject Classification. 42B25 (42B20).