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 \([W_{p,p}^1(I)]^*\).
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Funding
This work was supported in part by the Russian Foundation for Basic Research, project no. 19-01-00223.
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Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 312, pp. 236–250 https://doi.org/10.4213/tm4131.
Translated by I. Nikitin
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Prokhorov, D.V. On a Class of Functionals on a Weighted First-Order Sobolev Space on the Real Line. Proc. Steklov Inst. Math. 312, 226–240 (2021). https://doi.org/10.1134/S0081543821010144
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DOI: https://doi.org/10.1134/S0081543821010144