| Hardy-Littlewood maximal function is extremely important in the researching of harmonic functions, and one of the applications is weighted norm integral inequalities.In this paper, the principal problem considered is two weighted norm integral inequalities for the Hardy-Littlewood maximal function. the property of Ap -weight is generally introduced. And a full view of the theory of inserting weight to the equalities for the H-L maximal function is also given. Various related problems are considered, these include the relationship between H-L maximal operator Tf strong type ( p , p ) and the weight u ( x ), so does the weak type ( p , p ) and the weight u ( x ). A new result about weighted norm inequalities for H-L maximal function with Arλweight is given in chapter three. then introduced two weighted Hardy-Littlewood norm integral inequalities with Ap weight, given the relationship between H-L maximal operator Tf weak type ( p , p ) and the weights u ( x ), v ( x ). Based on this, several theorems about the relationship between Hardy-Littlewood operator Tf strong type ( p , p ) and the Ap weights were deduced out, including a suffeicent condition and a useful corollary. the main results were based on the theories of Calderón-Zygmund and Marcinkiewicz. Some results on norm integral inequalities with Ar,λ weight which is defined by S. Ding in 2000 are also given in the end.
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