Some $q$-analysis variants of Hardy type inequalities of the form $$ \int _0^b \bigg (x^{\alpha -1} \int _0^x t^{-\alpha } f(t) {\rm d}_q t \bigg )^{p} {\rm d}_q x \leq C \int _0^b f^p(t) {\rm d}_q t $$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.
In the paper we find conditions on the pair (ω1, ω2) which ensure the boundedness of the maximal operator and the Calderón-Zygmund singular integral operators from one generalized Morrey space Mp,ω1 to another Mp,ω2 , 1 < p < ∞, and from the space M1,ω1 to the weak space WM1,ω2 . As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.
The main purpose of this paper is to prove the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space.