$f+g\in L^p$ if $f,g \in L^p$

Asked Jun 24 '21 at 02:23

Active Jun 24 '21 at 02:23

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I want to show that $f+g \in L^p$ when both $f,g \in L^p$. The proof I'm give just says

Indeed $$|f(x) + g(x)|^p \le 2^p (|f(x)|^p + |g(x)|^p)$$ as can be seen by considering separately the cases $|f(x)| \le |g(x)| $ and $|g(x)| \le |f(x)|$.

Can anyone help me understand where $2^p$ came from and how the inequality holds?

asked Jun 24 '21 at 02:23

MoneyBall

4

if $|f(x)\leq|g(x)|$ then $|f(x)+g(x)|^p\leq (|f(x)|+|g(x)|)^p\leq (2g(x))^p=2^p|g(x)|^p$. Similar thing when $|g(x)\leq |f(x)|$. So $|f(x)+g(x)|^p \leq 2^p \max(|f(x)|, |g(x)|)^p \leq 2^p (|f(x)|^p+|g(x)|^p)$. – Stefan Lafon Jun 24 '21 at 02:30
3

does this answer your question ? – C Squared Jun 24 '21 at 02:31
Ah yes, thank you so much! – MoneyBall Jun 24 '21 at 02:40

$f+g\in L^p$ if $f,g \in L^p$

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