Higher-dimensional Knopp Type Inequalities And Hardy Type Q-inequalities

The sharp constants for the Hardy inequality and its various generalizations play an important role in harmonic analysis.In this thesis,we focus on two questions on Hardy inequalities:the characterization of the higher-dimensional weighted Knopp type inequality on the product space;the sharp constant for Hardy type q-inequality.This thesis is divided into three chapters.In Chapter 1,we mainly present the background of Knopp inequalities and Hardy q-calculus.In Chapter 2,we give a necessary and sufficient condition on weight pairs for a class of higher-dimensional weighted Knopp type inequalities to hold.The corresponding result for the multivariate adjoint Hardy type operator is obtained.The generalized result for the higher-dimensional weighted Knopp type inequality is also discussed.In Chapter 3,we focus on the two-dimensional Hardy operator in q-analysis of the form H2(f)(x1,x2)=1/x1x2?0x1?0x2?(y1,y2)dqy1dqy2,x1,x2>0.The sharp constant for the two-dimensional Hardy inequality on Lebesgue spaces with power weights is calculated.Similarly,the best constant for the bilinear Hilbert inequality in q-analysis on Lebesgue spaces with power weights is also obtained.