Lipschitz Continuous Functions on C([a,b])

Question

Hi Guys, im kinda stuck on a problem which states:

Let $g_0 \in C^0([a,b])$. Prove that $\forall p\in [1,\infty)$ the function $\varphi:C{_p}^0([a,b]) \to \mathbb{R}$ given by:

$\varphi(f)=\int_{a}^{b} fg_0$

Is Lipschitz Continuous

So as far as i can go, i tried to use the Holder Inequality for this:

Let $1<p<\infty$ and let $f$,$h$ $\in C_{0}^p([a,b])$. And by Holder's inequality we can obtain that $q=$$p\over (p-1)$ and applying it we get:

$|\int_{a}^{b}(f-h)g_0|\leq \int_{a}^{b}|f-h||g_o|\leq M_0(b-a)^{1/q}(\int_{a}^{b}|f-h)|^p)^{1/p}$

Where $M_0$ is an upper bound of $|g_0|$

But what happens when $p=\infty$?

I think that the Lipschitz constant it's going to be a kind of $max$ that lies in [a,b], but i get confused. Any tip or correction would be helpful.

Answer 1

$|\int (f-h) g_0| \leq \|f-h\|_{\infty} \int |g_0|$. So $\int |g_0|$ is a Lipschitz constant.