The Spherical Mean Value Operators on Euclidean and Hyperbolic Spaces. - Lim, Kyung-Taek. (2017)
Abstract: On Euclidean space, we prove the surjectivity of the spherical mean value operator from the class of smooth functions to itself, and from the class of distributions to itself, and obtain range characterizations of this operator on the class of compactly supported distributions and functions, respectively. On the three dimensional hyperbolic space, we obtain a range characterization of the spherical mean value operator on the class of compactly supported distributions. From this we show that this operator is surjective from the space of smooth functions onto itself. We extend some results on the spherical mean obtained by Fritz John. We derive a formula for the iterated spherical mean in hyperbolic spaces. We also show that a smooth function f on the three dimensional hyperbolic space with known averages over all spheres of a fixed radius r > 0 is uniquely determined, if the values of f are known on certain split annuli where the sum of the thicknesses of the annuli is r. Finally we obtain an explicit solution to the inhomogeneous spherical mean value equation in three dimensional hyperbolic space. We provide supplementary support theorems for the single radius spherical mean in Euclidean space of dimension 3 and 5, and a support theorem for the single radius spherical mean in Hyperbolic space of dimension 3.; Thesis (Ph.D.)--Tufts University, 2012.; Submitted to the Dept. of Mathematics.; Advisor: Fulton Gonzalez.; Committee: Sigurdur Helgason, Eric Todd Quinto, and Loring Tu.; Keyword: Mathematics.