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My question is analogous to the embedding $L^p\subset L^q(\Omega)$, for $p>q$ and for a bounded $\Omega$. In weak $L^p$ spaces, that is, $L^{p,\infty}$, does such an inclusion hold for arbitrary sets, where

$|||f|||_{L^{p,\infty}}^p=\sup_{\lambda>0}\left\{\lambda^p\mathcal{L}^N\left\{|f|>\lambda\right\}\right\}$

I would hope that an estimate like this would hold: $|||f|||_{L^{p,\infty}}\leq |||f|||_{L^{q,\infty}} $ for $p<q$. I have no counterexample. This is a little tricky because it defines a pseudonorm, not a norm.

Thank you to whoever answers.

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If $\lambda\le1$ then $|\{|f|>\lambda\}|\le|\Omega|$. If $\lambda>1$ then $$ |\{|f|>\lambda\}|\le\frac{\|f\|_{p,\infty}}{\lambda^p}\le\frac{\|f\|_{p,\infty}}{\lambda^q}. $$

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  • $\begingroup$ Thanks, I feel like a idiot. :) $\endgroup$ – michek Feb 4 '14 at 9:58
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    $\begingroup$ Please don't. We all go through situations like this. $\endgroup$ – Julián Aguirre Feb 4 '14 at 12:57
  • $\begingroup$ @Aguirre is there a documents where i can found all the embeddings??? $\endgroup$ – Poline Sandra Nov 27 '18 at 16:34

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