Generalized Littlewood-Paley Characterizations Of Sobolev Spaces

In this paper,we characterizes the f'ractional Sobolev space W?,p(Rn)and weighted Sobolev space Ww?,P(Rn)via the generalized Littlewood-Paley square function containing the measure ? that satisfies the Ahlfors regularity.This result generalizes the generalized Littlewood-Paley square function with compact support to the general measure ?,and characterizes the relationship between the Ahlfors regularity of the measure and the regularity indicator ? of Sobolev space.It also eliminates the limitations of compact support conditions.For weighted Sobolev space Ww?,p(Rn),we obtained corresponding results using similar methods,and improved the latest resultsThe paper is divided into three chaptersThe first chapter introduces the background knowledge of Sobolev space W?,pP(Rn)and weighted Sobolev space Ww?,p(Rn),some equivalent descriptions and developments of their definitions,the present situation of the research and the main content of this paperIn chapter two,we recall the definition of Littlewood-Paley square function,prove the result of generalized Littlewood-Paley square function characterizes the fractional Sobolev space W?,p(Rn).The corresponding background knowledge and the proof of the main theorem are givenIn chapter three,We recall the definition of Ap weights,present the result and proof of generalized Littlewood-Paley square function characterizes the weighted Sobolev space Ww?,p(Rn).