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)| $
Can we 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'm not sure if this is related to the the Amalgam space $W(L^{\infty},L^{2})$ !!)
I don't know exactly the next step for $\int_{-\infty}^{\infty}|f^{\#}(x)|^{2}dx=\int_{-\infty}^{\infty}\big|\sup_{y\in[x-1,x+1]}|f(y)| \big|^{2}dx$