Mixed Radial-angular Integrabilities For Rough Operators And Related Problems

The research on the boundedness of singular integrals and related operators in various types of function spaces is one of the core topics in the field of harmonic analysis.Classical research methods generally depend on the smoothness of the kernel function.Weakening or removing the smoothness of the kernel function,whether the boundedness of the operator is still valid or not has always been a hot topic in harmonic analysis.This dissertation aims to study on the mixed radial-angular integrabilities of singular integrals with rough kernels and related operators and the boundedness of such operators on Lebesgue and weighted Lebesgue spaces.Firstly,we focus on the boundedness of singular integrals with the rough kernels and the corresponding truncated maximal operators on the mixed radial-angular spaces Lradp Langp(Rn)under the sphere kernel function Ω∈ H1(Sn-1).Secondly,we pay attention to study the boundedness of parabolic singular integrals with rough kernels,the corresponding truncated maximal operators and parabolic Marcinkiewicz integral operators with rough kernels on the mixed radial-angular spaces LradpLang p(Rn)under the GrafakosStefanov condition Gβ(Sn-1)(β>1),respectively.Furthermore,when the kernel functions satisfy the weaker Grafakos-Stefanov condition WGβ(Sn-1)(β>1)on the unit sphere and the certain rather weakened radial size condition,we establish the mixed radial-angular integrabilities of singular integrals with rough kernels along the polynomial curves and the corresponding truncated maximal operators.These conclusions are not only the generalizations of the classical Lp-boundedness,but also improvements on previous results,where the spherical size conditions of the kernel function satisfying Ω∈ H1(Sn-1)and Gβ(Sn-1)(β>1)are’t proper subsets of each other,which are the weakest at present in the theory of rough kernel singular integral operators.In addition,for singular integral operators with rough kernels associated to surfaces and the corresponding truncated maximal operators,when the kernel functions Ω∈ WGβ(Sn-1)(β>1)and h ∈Δγ(R+)(γ∈(1,∞]),the weighted Lp-boundedness of such operators are established.Finally,when the kernel functions Ω∈Lq(Sn-1)(q ∈(1,2])and h ∈ Δγ(R+)(γ∈(1,∞]),we establish the Lp-boundedness of parametric Marcinkiewicz integral operators with rough kernels along real-analytic submanifolds.These conclusions obtained are the generalizations and improvements of some classical results.