Continuity of point evaluation on space of Hölder functions with $L^p$ norm

Let $$L>0$$ and $$\Omega \subset \mathbb{R}^n$$ a bounded Lipschitz domain. Define $$B_{\frac12,L}:=\{f \in L^2((0,1) \times \Omega) : \|f(t,\cdot)-f(s,\cdot)\|_{L^2(\Omega)} \leq L|t-s|^{\frac12},~ \forall s,t \in [0,1]\}.$$ I would like to show that, for every fixed $$s,t \in [0,1]$$, the functional $$f \mapsto \int_\Omega |f(t,x)-f(s,x)|$$ is continuous with respect to $$L^2((0,1) \times \Omega)$$-norm on the set $$B_{\frac12,L}$$, . It seems to me that this amounts to the question of continuity of point evaluation on a set of Hölder continuous functions with respect to $$L^2$$-norm, but I was not able to show it.

Does anyone have any direction: reference, counterexample, proof (hopefully)?

• I have troubles parsing what your actual question, or rather the actual functional, is. Do you want $f \mapsto \sup_{t,s \in [0,1]} \int_\Omega |f(t,x)-f(s,x)| \, \mathrm{d}x$ continuous with respect to the $L^2(I \times \Omega)$ norm on $B_{1/2,L}$? Aug 12, 2021 at 8:48
• I edited the question to clarify a bit. So, for every fixed $s,t$ you want that $|Af-Ag|$ is small for small $\|f-g\|_{L^2(I\times \Omega)}$, where $A$ is the operator defined above. Aug 12, 2021 at 9:35

I believe that I found a way to solve the problem. Let us show that the point evaluation $$f \mapsto f_t$$ is a bounded operator from $$L^2((0,1) \times \Omega)$$ to $$L^2(\Omega)$$ and the rest follows analogously. Namely, from the given condition, for every $$s \in [0,1]$$ we have $$\|f_t\|_{L^2(\Omega)}^2 \leq 2(L^2|t-s| + \|f_s\|_{L^2(\Omega)}^2)$$ and thus $$\sup_{\|f\|_{L^2((0,1)\times \Omega)}=1} \|f_t\|_{L^2(\Omega)} \leq \sqrt{2L^2+2}$$, for every $$t \in [0,1]$$, from where the boundedness follows.