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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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    Ok, I nearly see it now. Except for the mean-value theorem step...2012-09-19

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Just to put something in this box: the "mean-value theorem step" is

For all real $r\ge 1$ one has $r^{p-1} - 1 \leq c_p(r - 1)(r^{p-1} + 1)$

Indeed, applying MVT to $f(x)=x^{p-1}$ on the interval $[1,r]$ we get $f(r)-f(1)=f'(\xi)(r-1)=(p-1)\xi^{p-2}(r-1),\qquad \exists \xi\in (1,r)$ Here $\xi^{p-2}\le \max(r^{p-2},1)$ where we take $\max $ because $p-2$ could be either negative or positive. Hence, $r^{p-1} - 1 \leq (p-1)(r-1) \max(r^{p-2},1)\leq (p-1)(r-1) (r^{p-1}+1)$