Mapping Properties of Certain Averaging Operators

Citation

Erdoğan, Mehmet Burak (2002) Mapping Properties of Certain Averaging Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JRFS-4S52. https://resolver.caltech.edu/CaltechTHESIS:01242012-162804546

Abstract

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.

Thesis Files