Abstract
We investigate natural sufficient conditions for a space \(L^p(m)\) of \(p\)-integrable functions with respect to a positive vector measure to be smooth. Under some assumptions on the representation of the dual space of such a space, we prove that this is the case for instance if the Banach space where the vector measure takes its values is smooth. We give also some examples and show some applications of our results for determining norm attaining elements for operators between two spaces \(L^p(m_1)\) and \(L^q(m_2)\) of positive vector measures \(m_1\) and \(m_2\).
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The authors would like to thank the anonymous referee for his/her comments that helped us to improve the paper.
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Communicated by G. Teschl.
Professor Agud and professor Sánchez-Pérez authors gratefully acknowledge the support of the Ministerio de Economía y Competitividad (Spain), under project #MTM2012-36740-c02-02. Professor Calabuig gratefully acknowledges the support of the Ministerio de Economía y Competitividad (Spain), under project #MTM2011-23164.
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Agud, L., Calabuig, J.M. & Sánchez-Pérez, E.A. On the smoothness of \(L^p\) of a positive vector measure. Monatsh Math 178, 329–343 (2015). https://doi.org/10.1007/s00605-014-0666-7
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DOI: https://doi.org/10.1007/s00605-014-0666-7