Boundedness of discrete Hardy-type operators and self-improving properties of discrete Ariňo and Muckenhoupt weights

In this paper, we characterize the boundedness of a generalized Hardy operator and its adjoint on $$\ell _{w}^{p}(\mathbb {Z}_{+})$$ , via the discrete weights w belonging to the discrete classes $${\mathcal {B}}_{p}^{\lambda }$$ and $${\mathcal {B}}_{p}^{*\lambda }$$ , respectively. As an application of the corresponding boundedness, we show that the self-improving property of a discrete class $${\mathcal {B}}_{p}$$ is valid, i.e., we prove that if $$w\in {\mathcal {B}}_{p}$$ for some $$10$$ such that $$w\in {\mathcal {B}}_{p-\varepsilon }$$ . Similarly, we also prove the self-improving property of a discrete class $${\mathcal {B}}_{p}^{*}$$ , i.e., we show that if $$w\in {\mathcal {B}}_{p}^{*}$$ for some $$10$$ such that $$w\in {\mathcal {B}}_{p+\varepsilon }^{*}.$$ Finally, the established results are also applied for proving the boundedness of the Hardy–Littlewood maximal operator on Lorentz sequence spaces.