Carleson Measure On The Unit Ball Of Rⁿ⁺¹ And Its Applications

The goal of this thesis is to study the properties of Carleson measures on the unitball B of Rn+1, including the defnition, the characterizations in terms of Lp-integration,the duality and their applications into operator theorems and function spaces defned onthe unit sphere.In chapter1, frstly, we mainly review some works and results concerning with Car-leson measures on the unit disk of the complex plane. Secondly, we introduce our aimsand main results, as well as the content distribution in this paper.In Chapter2, we introduce a new defnition for Carleson measures on the unit ballof Rn+1, which are defned in terms of Carleson boxes or tents. Then, we characterize theCarleson measures providing an integral form which is involved with the self-conformalmap on the unit ball. We also introduce the Lpintegration concerning with the derivativeof the self-conformal map on the unit ball to characterize the Carleson measures on theunit ball of Rn+1. Finally, by the Chouqet integral with respect to Hausdorf capacity,the duality space for Carleson measures on the unit ball of Rn+1is resolved.In Chapter3, as an application of Carleson measures on the unit ball of Rn+1, westudy a certain generalized area operator defned on the unit ball. Firstly, we defne anarea integral operator Aμwith the nonnegative measure μ on the unit ball. Then, as thepreliminaries, we establish the connections among maximal function, Poisson integral,the non-tangential maximal function and Carleson measures on the unit ball. Finally, wecharacterize the nonnegative measure μ such that Aμis bounded from Lpto Lqor fromHpto Lqby Carleson measures on the unit ball and other forms.In Chapter4, as another application of Carleson measures on the unit ball of Rn+1,we study the function spaces Qα(Sn) defned on the unit sphere. According to the studyof Q spaces on the unit disk D of the complex plane and in the upper half space Rn+1, we defne Qnα(S) spaces on the unit sphere Snfrstly. Then, we study the basic properties ofQα(Sn) spaces, which keep consistently with the situations on the unit disk. At last, wecharacterize Qnα(S) spaces by Carleson measures on the unit ball of Rn+1.In Chapter5, the third application of Carleson measures on the unit ball of Rn+1is to study Morrey spaces on the unit sphere. Since the closed relations among Carlesonmeasures, Q spaces and Morrey spaces, we study Morrey spaces defned on the unit sphereSnand characterize Morrey spaces by Carleson measures on the unit ball.