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I am trying to prove the estimate for $a,b\in \mathbb{R^n}, p\geq 1$ $$|a+b|^p\leq C(|a|^p+|b|^p)$$ where $C$ is a constant depending only on $p$.

I am able to do it when $p$ is a natural number. But, could you give me any hints on how to do it for any $p\geq 1$?

Dimension Entangled
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    See https://math.stackexchange.com/q/3196836/42969, $C=2^{p-1}$ is the best constant. – Martin R Jun 15 '21 at 18:55
  • Dividing by $|a|^p$ you need the inequality: $$|1+x|^p\leq C(1+|x|^p)$$ Not sure if that helps. You can pick $|a|\leq |b|$ so $|x|\geq 1.$ Trying $x=1,$ this shows that you need $C\geq 2^{p-1}.$ – Thomas Andrews Jun 15 '21 at 18:58
  • @ThomasAndrews And in fact $C\ge 2^{p-1}$ is sufficient, as $\frac{(1+|x|)^p}{1+|x|^p} $ attains its maximum at $x=1$, and $|1+x |^p \le (1+|x| )^p$. – Vishu Jun 15 '21 at 19:12
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    https://math.stackexchange.com/q/143173/321264 – StubbornAtom Jun 15 '21 at 19:37

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