I can't find if this is true ( Lebesgue norms)

Question

If $p,q\in (1,\infty)$ and $p \leq q$ and $u\in L^q(\Omega)$ then it is true that $||u||_{L^q(\Omega)} \leq ||u||_{L^p(\Omega)}$ ?

P.S. $\Omega$ is open, bounded and $\partial \Omega \in C^\infty (\Omega)$

Answer 1

Let $m$ denote Lebesgue measure. Since $\Omega$ is bounded and open, $0<m(\Omega)<\infty$, so Jensen's inequality yields $$\left(\frac{1}{m(\Omega)}\int_{\Omega}|u|^p\ dm\right)^{q/p}\leq\frac{1}{m(\Omega)}\int_{\Omega}|u|^q\ dm,$$ and thus $$\|u\|_{L^p(\Omega}\leq m(\Omega)^{\frac{1}{p}-\frac{1}{q}}\|u\|_{L^q(\Omega)}.$$