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Let $\alpha,\beta$ be two multi-indices. Let $\Omega$ be a domain in $\mathbb{R}^n$. Assume $u\in L^1_{loc}(\Omega)$.

We know that if weak derivative $D^{\alpha+\beta}u$ exists, then $D^\alpha u$ or $D^\beta u$ may NOT exist. And if weak derivatives $D^\alpha u,D^\beta u$ exist, then $D^{\alpha+\beta}u$ may NOT exist.

The question is that if $D^{\alpha+\beta}u$ and $D^{\beta}u$ exist, then can we have $D^\alpha u$ exists?

yangmengqh
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  • I can't make any sense of this. First, how can $D^{\alpha+\beta}$ exist if $D^\alpha$ does not exist? Second, the hypothesis in the question assumes something you say we know cannot happen... – David C. Ullrich Aug 12 '16 at 13:12
  • These derivatives are all in weak sense. It can happen that some higher order derivative exists without existence of lower order derivatives. – yangmengqh Aug 12 '16 at 13:16
  • What's an example where that happens? (Even if that can happen, it's still true that you say we know A cannot happen and then your question begins by assuming A happens - ???) – David C. Ullrich Aug 12 '16 at 13:18
  • The question is no longer self-contradictory after the edit. I'd still like an example where $D^{\alpha+\beta}$ exists but $D^\alpha$ does not. You say we know this is possible - I don't know any such thing. – David C. Ullrich Aug 12 '16 at 13:23
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    http://math.stackexchange.com/questions/1141768/does-the-existence-of-weak-derivatives-require-the-lower-order-derivatives-also – yangmengqh Aug 12 '16 at 13:24
  • Thank you! I have modified this problem. – yangmengqh Aug 12 '16 at 13:26

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