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The book proves the following $u(t)=u(s)+\int_{s}^{t}u'(\tau)d\tau$ and says this equality implies "easily" that $\max_{0\leq t\leq T}\lVert u(t)\rVert\leq C \lVert u\rVert_{W^{1,p}(0,T;X)}$ where $X$ is a Banach space and $||u||_{W^{1,p}(0,T;X)}^p=\int_{0}^T \Vert u(t)\Vert ^p+\Vert u'(t)\Vert ^p dt$, and $C$ is a constant depending on $T$.

I do not know how to insert the exponent $p$. I was looking into the Bochner Integral properties but I can not figure it out.

I am trying to show $\Vert u(t_0)\Vert\leq C\int_{0}^T \Vert u(t)\Vert^p+\Vert u'(t)\Vert ^p dt$ but I do not know how to insert the $p$. Please help me.

Or where can I find exercises about how to do this estimates? Is there an easier book?

K.defaoite
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    These are the steps: take norms, estimate norm of integral by integral of norm, extend integral to $(0,T)$, raise to power $p$, apply $(a+b)^p \le 2^{p-1} (a^p+b^p)$, Hoelder inequality for the integral. Then integrate the right-hand side wrt $s$ <- this is the main trick – daw May 10 '25 at 12:49
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    here is a related question https://math.stackexchange.com/questions/4763660/brezis-theorem-8-8-how-to-get-the-inequality-u-l-inftyi-leq-c-u?rq=1 – daw May 10 '25 at 12:53

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