Some Generalizations And Applications Of Hardy Inequality

In this paper,we give some generalizations of Hardy Inequality on various aspects and their related applications.In the first chapter,we proceed from the most generalized form of Hardy Inequality,re-examine a conclusion of Boyd's,broaden the range of the inequality established,give the equal conditions,and find its dual inequality.In the second chapter,we make a comprehensive improvement and promotion of J.Bergh's discrete and integral Hardy Inequality for the monotonic situations,which lead to a more complete system of this kind of Hardy Inequality.In Chapter 3,we introduce a kernel function and view the parts of the negative exponential Hardy Inequalities on a unified point of view,and find a new negative exponent Hardy Inequality.This chapter also solves a discrete negative exponent Hardy Inequality optimal constant problem which proposed by Yang Bicheng.In Chapter 4,we re-examine two generalized forms of Carleman's Inequalities,discuss their correctness and optimal constants in detail,and get a correct and relatively complete conclusion.In Chapter 5,we completely solve a class of mixed norm Hardy Inequality problems,and we determine the correct optimal constants and equal conditions.We determine the differential equation whose optimal solution is derived through the extreme value theory.On the other hand,by solving a partial solution of a nonlinear differential equation,we guess the best constant and optimal solution,furthermore,the resulting inequality in the continuous case is established by carefully constructing the auxiliary functions.Finally,this result is extended to the measurable functions by the real analysis theory.In Chapter 6,we extend the basic generalized form of the Hardy Inequality of one variable to the multivariate case,and as an application,we make use of the related Hardy Inequality to judge the existence and uniqueness of the generalized solution of the stationary NS equation.