Research On Several Weighted Hardy-Rellich Type Inequalities

Hardy type inequality has important applications in physics and many branch of mathematics.In order to develop the mathematics itself and solve the problem of physics and other related disciplines,it is necessary to carry out Hardy type inequality all aspects of the research and extension.Hardy-Rellich type inequality as Hardy inequality in higher derivative of promotion,also caught the attention of many scholars.Based on harmonic analysis tools,there are more in-depth research for several kinds of weighted Hardy-Rellich inequalities,and the problem of the best constant is taken into account.In this paper,the main results are as follows:1.Using spherical harmonic decomposition technique,there is a class of weighted Hardy-Rellich inequality in three and more dimension,and the optimality of the constant is proved.2.Through the h-harmonic decomposition technique,we get a class of the weighted Hardy-Rellich inequality based on Dunkl operator,and prove the constant is the best.3.Beacause of the careful analysis,we first get a class of CKN inequality based on Finsler metrics,and then,we obtain a class of the weighted Hardy-Rellich inequality based on Finsler metrics.In this paper,using the analysis of common technology,the developement of a few class of Hardy type inequality is promoted,especially,we get several types of the optimal weighted Hardy-Rellich inequality,also complete and develope the study of Hardy inequality.