Inégalités de Hardy et applications

Abstract

About the year thirty, an inequality called Hardy’s inequality was proved [See {45}, {48} ] : for and an arbitrary measurable positive function we have . (IH) with condition . is an optimal constant (the less possible). If then (IH) is not true for arbitrary measurable positives functions , but it is for non-negative non-increasing functions with optimal constant [See{23}]. In {87} inequality of type (IH) was proved under weaker assumptions on f but still of monotonicity type. The aim of this thesis is : 1) To establish the result of {87} differently which enable us to get a new constant which is optimal. 2) We prove an analogue result for a generalized Hardy operator. An optimal constant is obtained and we specify the function for which we get equality in (IH). 3) We obtain conditions ensuring the validity of an analogue of inequality (IH) for the weighted Lebesgue spaces and . In a second part of this thesis we give some examples of applications of Hardy’s type inequalities. Then we establish two Hardy’s type inequalities which we use to prove the equivalence of norms ( ) and quasi-norms ( ) of Besov spaces defined via modulus of continuity and defined via differences, with , , and two parameters of regularities [the classical parameter l and a slowly varying function ].