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I recently came across the following notation in a paper. The author first estimated \begin{equation} \vert u (1-\vert u \vert^2)\vert \leq 2 \left[ \left\vert 1-\vert u \vert^2 \right\vert 1_{\lbrace \vert u \vert \leq 2\rbrace} + \left( \vert u \vert^2-1 \right)^{\frac{3}{2}} 1_{\lbrace \vert u \vert \geq 2 \rbrace} \right] \end{equation} which is correct.

Then he goes on to say

\begin{equation} \Vert u(1-\vert u \vert^2) \Vert_{L^2+L^{\frac{4}{3}}} \leq 2 \left[ \Vert 1 - \vert u \vert^2 \Vert_{L^2}+ \Vert 1 - \vert u \vert^2 \Vert_{L^2}^{\frac{3}{2}} \right] \end{equation}

My question is:

What is the "$L^2+L^{\frac{4}{3}}$-Norm" supposed to be in this scenario?

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The usual definition of such a norm would be $$ \Vert u \Vert_{L^2+L^{\frac{4}{3}}} =\inf\{\|u_1\|_{L^2}+\|u_2\|_{L^{4/3}}:u=u_1+u_2,u_1\in L^2,u_2\in L^{4/3}\}. $$

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  • $\begingroup$ Thanks a lot! That also enables me to do the second estimate in my post. $\endgroup$ – mjb Jun 12 '13 at 7:21

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