In this paper, we prove some new characterizations of weighted functions for dynamic inequalities of Hardy’s type involving monotonic functions on a time scale \(\mathbb {T}\) in different spaces \(L^{p}(\mathbb {T})\) and \(L^{q}( \mathbb {T})\) when \(0<p<q<\infty \) and \(p\le 1\). The main results will be proved by employing the reverse Hölder inequality, integration by parts, and the Fubini theorem on time scales. The main contribution in this paper is the new proof in the case when \(p<1\), which has not been considered before on time scales. Moreover, the results unify and extend continuous and discrete systems under one theory.

在本文中，我们证明了在时间尺度\（\ mathbb {T} \）在不同空间\（L ^ {p}（\ mathbb {T} ）\）和\（L ^ {q}（\ mathbb {T}）\）当\（0 <p <q <\ infty \）和\（p \ le 1 \）时。主要的结果将通过使用逆Hölder不等式，部分积分和时标上的Fubini定理来证明。本文的主要贡献是在\（p <1 \）的情况下的新证明，这在时间标度之前尚未被考虑。而且，结果在一种理论下统一并扩展了连续和离散的系统。