I am trying to prove the following statement
For $1<q\leq p \leq \infty$ it holds
$\|S(t) w \|_{L^p} \leq t^{\frac{1}{p}-\frac{1}{q}} \|w\|_{L^q}$ for all $t>0$
and for each $w \in L^q(\Omega)$
where $S(t)$ is an analityc semigroup and $\Omega \subset R^{n}$ is open and bounded.
So far I have proved my claim in this cases:
i) $1<q \leq p < \infty$
ii) $1<q<p \leq \infty$
So to finish the proof I only lack the case $p=q=\infty$
I want to know if my idea to finish the test is correct or else can you suggest me how to complete the demonstration?
My idea is this, proof $p=q=\infty$, Let $w \in L^{\infty}(\Omega)$ then $w \in L^q$ for all $q<\infty$ and apply case i) we get
$\|S(t) w \|_{L^q} \leq \|w\|_{L^q}$
Then taking the limit when p tends to infinity, we arrive to
$\|S(t) w \|_{L^{\infty}} \leq \|w\|_{L^{\infty}}$ this is correct? I assume that $\lim_{p \to \infty} \|w\|_p=\|w\|_{\infty}$
Thanks for your help and your comments I have been improving the post and sorry for my english.