Abstract

Let \(g\) be a Lebesgue measurable function on an interval \(I\subset\mathbb R\). We find conditions on \(g\) under which the mapping \(f\mapsto \intop_I g(x)(Df)(x)\,dx\) is a continuous linear functional on a weighted first-order Sobolev space \(W_{p,p}^1(I)\); we also obtain estimates for the norm of this functional in \([W_{p,p}^1(I)]^*\).

中文翻译：

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关于实线上加权一阶Sobolev空间上的一类泛函

摘要

令\（g \）为区间\（I \ subset \ mathbb R \）上的Lebesgue可测量函数。我们在\（g \）上找到条件，在该条件下映射\（f \ mapsto \ intop_I g（x）（Df）（x）\，dx \）是加权一阶Sobolev空间\ { W_ {p，p} ^ 1（I）\） ; 我们还在\（[W_ {p，p} ^ 1（I）] ^ * \）中获得此函数范数的估计。