AI 中文总结
本文在加权变量Lebesgue空间中，为广义积分算子建立了模不等式成立的充要权条件，并给出模不等式导出范数不等式的条件，最后通过例子验证结果。
AI 中文摘要
我们建立了一个必要且充分的权条件，使得广义积分算子$T_\phi$在加权变量Lebesgue空间$L_w^{p(x)}$中的非负非增函数上满足模不等式。我们给出了一个条件，在该条件下，所提及的模不等式可导出范数不等式。最后，给出例子以说明我们的主要结果，并为一些著名算子获得模不等式。
英文摘要
We establish a necessary and sufficient weight condition so that a modular inequality holds for a generalized integral operator $T_ϕ$ on non-negative non-increasing functions in weighted variable Lebesgue spaces $L_w^{p(x)}.$ We give a condition under which the mentioned modular inequality leads to a norm inequality. Lastly, examples are given to demonstrate our main result, to obtain modular inequalities for some well known operators.
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