So, I'm posting this question again because there is new given information for the problem, and I think people just answering new posted questions:
Let $f(x)$ be a continuous and locally bounded function on $\mathbb R$, then the local maximum function is defined by $ f^{\#}(x)=\sup_{y\in[x-1,x+1]}|f(y)| $ Edit: We can replace the "$1$" in the "sup" by any nember $d\leq 1$.
I'm trying to find a relation between the $L^{2}$ norm of $f^{\#}$ and the $L^{2}$ norm of $f$ ? (if we know that $\|f\|_{L^{2}(\mathbb R)}<\infty$) i.e., something like $\|f^{\#}\|_{L^{2}(\mathbb R)}\leq C\, \|f\|_{L^{2}(\mathbb R)}$.
The fears in the prevous answers were that $f$ could have (sharp) peaks, which makes $f^{\#}\notin L^{2}(\mathbb R)$.
Now, given the above information about $f$, and assuming that $f$ is continuously differentiable, with bounded derivaive on $\mathbb R$, and that $f$ is bounded on $\mathbb R$, does this change the argument to have the result which I'm looking for!
Link to old post: Local maximum function