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For $u,v \in L^q(\Omega)$ with $q \ge p \ge 1$, how does one show that: $$ \begin{aligned} \||u|^{p-1}u - |v|^{p-1}v\|_{L^{p/q}} & \le C\,\|(|u|^{p-1} + |v|^{p-1})\,|u-v|\,\|_{L^{p/q}}\\ & \le C\,(\|u\|^{p-1}_{L^q} + \|v\|^{p-1}_{L^q})\,\|u-v\|_{L^q} \end{aligned} $$

Thanks.

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    the first inequality is well explained here http://math.stackexchange.com/questions/9960/complex-inequality-up-1u-vp-1v-leq-c-p-u-vup-1vp-1?rq=12012-09-19
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    ... and the second one is Hölder with $\frac 1q + \frac 1\alpha = \frac 1{\frac pq}$ giving $\alpha = \frac pq + \frac 1{p-1}$.2012-09-19
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    Ok, I nearly see it now. Except for the mean-value theorem step...2012-09-19

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