Mapping Properties of Certain Averaging Operators

Author: Erdoğan, Mehmet Burak

Year: 2002

Degree: Dissertation (Ph.D.)

Advisors: Wolff, Thomas H.; Ramakrishnan, Dinakar

Committee Members: Makarov, Nikolai G.; Hundertmark, Dirk; Simon, Barry M.; Wales, David B.; Wolff, Thomas H.; Ramakrishnan, Dinakar

Option: Mathematics

DOI: 10.7907/JRFS-4S52

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.

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