In this thesis, we investigate the mapping properties of two averaging operators.
In the first part, we consider a model rigid well-curved line complex G_d in R^d. The X-ray transform, X, restricted to G_d is defined as an operator from functions on R^d to functions on G_d in the following way:Xf(l) = ∫_lf,l ϵ G_d.We obtain sharp mixed norm estimates for X in R^4 and R^5.
In the second part, we consider the elliptic maximal function M. Let ε be the set of all ellipses in R^2 centered at the origin with axial lengths in [1/2,2].Let f : R^2 -> R, then M f : R^2 -> R is defined in the following way:Mf(x) = ^(sup)_(Eϵε) ^1/_(|E|) ∫_E f(x+s)dσ(s),where dσ is the arclength measure on E and |E| is the length of E.
In this part of the thesis, we investigate the L^P mapping properties of M.
