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Let $p\in ]0,1]$ and $f,g$ be two measurable functions. For $l\in \Bbb R, l\gt 0$ and a measurable function $k$, $N_l (k)$ is defined as follows: $$N_l(k) := (\int_X \mid k\mid ^l d\mu)^{1/l}$$

Now I have to show the following two properties of $N$:

  1. $N_p^p (f+g) \le N_p^p(f) + N_p^p (g)$
  2. $N_p(f+g)\le 2^{\frac{1}{p} -1} (N_p(f)+N_p(g))$

Any tipps or ideas on how to solve this? Thanks in advance !

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    are you sure that $l>0$ instead of $l\geq 1$? since for your first question this would be the minkowski inequality in that case. – 2017-01-08
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    yes i know, but p has to be between 0 and 1. We've already proven the minkowski inequality, i.e. the case for p larger that 1 – 2017-01-08
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    what I can tell is that (at least for the first case) it would be enough to show that for simple functions, e.g. the case w.l.o.g. $k=\mathrm{const}$. – 2017-01-08
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    seems like there is all you need: https://matthewhr.files.wordpress.com/2012/09/lp-spaces-for-p-in-01.pdf – 2017-01-08

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