The Improvements Of Leray-Trudinger Inequality

Sobolev inequality,Hardy inequality and Moser-Trudinger inequality are several important inequalities,which are widely used in many fields such as mathematical physics.Based on the above inequalities,this thesis mainly discusses the improved form of Leray-Trudinger inequality,which is closely related to the above inequalities.The main conclusions of this thesis are as follows:(1)We obtain the series expansion of Leray-Trudinger inequality on the bounded region of Euclidean space.The proof process is based on the research results of Psaradakis-Spector and Mallick-Tintarev on Leray-Trudinger inequality,and adopts Trudinger’s proof method,i.e.L~q estimate.(2)We obtain the series expansion of Hardy inequality on the bounded region of Heisenberg Group and the best constant;we prove the Leray-Trudinger inequality on the bounded region of Heisenberg Group.On the basis of the first problem,a similar conclusion is considered on the Heisenberg group.Firstly,the series expansion of Hardy inequality and its optimal constant are obtained by vector field method.Secondly,the expansion of Leray-Trudinger inequality on the Heisenberg Group is obtained by L~q estimate method.