In this paper, we establish some conditions on nonnegative rd-continuous weight functions \(u\left( x\right) \) and \(\upsilon \left( x\right) \) which ensure that a reverse dynamic inequality of the form

$$\begin{aligned} \left( \int _{a}^{\infty }f^{p}(x)\upsilon \left( x\right) \Delta x\right) ^{ \frac{1}{p}}\le C\left( \int _{a}^{\infty }u\left( x\right) \left( \int _{a}^{\sigma \left( x\right) }\mathcal {K}\left( \sigma \left( x\right) ,\sigma \left( y\right) \right) f(y)\Delta y\right) ^{q}\Delta x\right) ^{ \frac{1}{q}}, \end{aligned}$$

holds when \(q\le p<0\) and \(0<q\le p<1.\) Corresponding dual results are also obtained. In particular, we prove some reverse dynamic weighted Hardy-type inequalities with kernels on time scales which as special cases contain some generalizations of integral and discrete inequalities due to Beesack and Heinig.

中文翻译：

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时间尺度上带有核的反向动态加权Hardy型不等式的刻画

在本文中，我们对非负rd连续权重函数\（u \ left（x \ right）\）和\（\ upsilon \ left（x \ right）\）建立一些条件，以确保形成

$$ \ begin {aligned} \ left（\ int _ {a} ^ {\ infty} f ^ {p}（x）\ upsilon \ left（x \ right）\ Delta x \ right）^ {\ frac {1 } {p}} \ le C \ left（\ int _ {a} ^ {\ infty} u \ left（x \ right）\ left（\ int _ {a} ^ {\ sigma \ left（x \ right） } \ mathcal {K} \ left（\ sigma \ left（x \ right），\ sigma \ left（y \ right）\ right）f（y）\ Delta y \ right）^ {q} \ Delta x \ right ）^ {\ frac {1} {q}}，\ end {aligned} $$

当\（q \ le p <0 \）和\（0 <q \ le p <1. \）时成立，也可以获得对应的对偶结果。特别是，我们证明了时间尺度上带有核的逆动态加权Hardy型不等式，在特殊情况下，由于Beesack和Heinig的存在，它们包含了积分和离散不等式的一些概括。