### A mixed norm estimate for the X-ray transform

Thomas Wolff
1998 Revista matemática iberoamericana
Let G be the space of lines in R 3 , i.e. the 4-dimensional manifold whose elements are all lines in R 3 . We can coordinatize G in the following way'='( e x) where e 2 S 2 nf 1g is the direction of'and x = x'is the unique point on'which is perpendicular to e. We will denote the direction e of'bỳ . The distance on G can bede ned using the standard distances on the sphere and in R 3 and this identi cation, thus d(' m) = jx'; x m j + (' m) where (' m) = (' m ) is the unoriented angle (2 0 = 2])
more » ... angle (2 0 = 2]) betweeǹ and m. This distance has the following property. Let T'(a) be the cylinder of radius , axis'and length 1, centered at the point a 2', and let T'= T'(x') where x'is as de ned above. Then for , (' m) and T'\ T m 6 = ? imply d(' m) C 0 where C 0 is a suitable numerical constant. All metric quantities de ned on G refer to the distance d. We will beusing mixed norms on G de ned in the following way: if F : G ;! R then kFk L q e (L r x ) def = Z e2S 2