Convergence Problems Of Certain Averaging Operators On Function Spaces

This dissertation is devoted to study convergence problems of certain av?eraging operators on function spaces.The contents of this dissertation can be divided into six chapters.The first chapter is intended to introduce the relevant background,the cur?rent research situation,main results of this dissertation,and then some prelimi?naries and auxiliary Lemmas.In Chapter 2,we investigate the convergence rate of the generalized Bochner-Riesz means BRδ,γ on Lp-Sobolev spaces on the Euclidean space Rn in the sharp range of δδ and p(p≥2).We give the relation between the smoothness im-posed on functions and the rate of almost-everywhere convergence of BRδ,r.As an application,the corresponding results can be extended to the n-torus Tn by using some transference theorems.Also,we consider the following two general-ized Bochner-Riesz multipliers,(1-|ξ|γ1)+δ and(1-|ξ|γ2)+δ,where γ1,γ2,δ are positive real numbers.We prove that,as the maximal operators of the multi-plier operators with respect to the two functions,their L2(|x|-β)-boundedness are equivalent for any γ1,γ2 and fixed δ.In Chapter 3,we give the rate of almost everywhere convergence of the combinations and multivariate averages on Sobolev type spaces on the Euclidean space Rn.The saturation of convergence is also obtained.When applied,we extend corresponding results to the n-torus Tn by means of transference theorems.In Chapter 4,we consider some multiplier operator μγ,α raised from studying the Lp-approximation of the spherical mean Stγ,and obtain the optimal range of exponents(α,γ,p)such that is an Lp multiplier.As an application,we obtain the convergence rate for Stγ(f)in the Lp spaces.In Chapter 5,we are concerned with the derivative operator of the general-ized spherical mean Stγ.By considering a more general multiplier and finding the smallest γ such that mγ,bΩis an Hp multiplier.We obtain the optimal range of exponents(γ,β,p)to ensure the Hp(Rn)boundedness of(?)βS1γf.As an application,we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on Hp(Rn)spaces.In Chapter 6,we study derivative estimates of the iterated spherical aver-ages(At)N(f).We obtain the optimal range of exponents(α,N,p)to ensure the Lp boundedness of P((?)/(?)x)(A1)N(f)for 1≤p≤∞,where P is a homogeneous polynomial of degree α.The main theorem extends some known results.As an application,we obtain the smallest N such that(A1)N:Lp(Rn)→Lαp(Rn).