Some Types Of Extensions Of Hardy’s Inequality

Starting from the background of Hardy’s inequality and the research status at home and abroad,this paper systematically studies the generalization proofs of several types of Hardy’s inequalities.The second chapter introduces the basic Hardy’s inequality and the preliminary knowledge used in this paper.The coefficient method and the divergence theorem study the proofs of two types of high-dimensional Hardy’s inequalities.In Chapter 4,we rigorously derive upper and lower bounds for the optimal coefficients of inequalities using the symmetric decreasing rearrangement theorem and a family of functions.In Chapter 5,we generalize Hardy’s inequalities to fractional order,and use the properties of specular transformation and hypergeometric functions to rigorously prove the optimal coefficients of such fractional Hardy’s inequalities.In Chapter 6 we obtain a class of Hardy-Sobolev inequalities that use specific functions to control the integral,and give the exact coefficients C.