Let $-\infty < a<b<\infty$ and $A \subseteq C([a,b]) \cap C^1((a,b))$ satisfying $\forall f\in A$ $\int_{b}^{a} |f(x)| dx + \int_{b}^{a} |f'(x)|^p dx < C$, where $C>0$ and $p>1$. I want to show uniform boundedness of functions from $A$.
It is easy to show using Hölder inequality that $\int_{b}^{a} |f(x)+f'(x) | dx < C'$ where $C'$ is some positive constant but I can't figure out uniform boundedness.
Use $f(x)-f(0) =\int_0^{x}f'(t) \, dt$ and Holder's inequality to show that $|f(x)-f(0)|$ is bounded, say $|f(x)-f(0)| \leq M$ for all $x$ for all $f\in A$. It is enough to show that $\{f(0):f \in A\}$ is bounded. Suppose, if possible, there exist $f_1,f_2,... \in A$ such that $f_n(0) \to +\infty$. Then $f_n(0) >C/(b-a)+M$ for $n$ sufficiently large. Hence $f_n(x) >C/(b-a)$ for all $x$ for $n$ sufficiently large. But then $\int_a^{b} |f_n(x)|\, dx >C$ for $n$ sufficiently large which is a contradiction. Similar argument holds if $f_n(0) \to -\infty$.