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Let \(u\in L^p({\Omega })\) with \(1\leqslant p<+\infty \) . Calculate \(\ell =\displaystyle \lim _{t\to +\infty }t^pm(t)\) . Show that if \(u\in L^{p,\infty }({\Omega }) \) then \(\sup _{t>0} t^p\big |\,|u|>t\big |<+\infty \) . Deduce that \(L^p({\Omega })\subset _{>} L^{p,\infty }({\Omega })\) by specifying the inclusion constant.

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Exercises and Problems

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  • 发表日期：

    2025年

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  • Jean Michel Rakotoson

摘要

Let \(u\in L^p({\Omega })\) with \(1\leqslant p<+\infty \) . Calculate \(\ell =\displaystyle \lim _{t\to +\infty }t^pm(t)\) . Show that if \(u\in L^{p,\infty }({\Omega }) \) then \(\sup _{t>0} t^p\big |\,|u|>t\big |<+\infty \) . Deduce that \(L^p({\Omega })\subset _{>} L^{p,\infty }({\Omega })\) by specifying the inclusion constant.