In this talk, we study the regularity property of averages over curves and associated maximal bounds.
First of all, we prove the optimal Lp Sobolev regularity estimate for averages over curves in R^d except some endpoint cases.
We settle the conjecture raised by Beltran, Guo, Hickman, and Seeger.
Secondly, we prove the local smoothing estimate of sharp order for averages over curves.
As a consequence, we establish, for the first time, Lp boundedness of the maximal averages over curves when d>3.
Lastly, we prove the maximal bound on the optimal range when d=3.