Let $w \in C_0^\infty(\mathbb{R},\mathbb{R}^+)$ be a function with $\int_\mathbb{R} w(x) = 1$.
What can we say about the first derivative, or what can we say about $\int_\mathbb{R} |\partial_xw|$ ?
I am especially interested in the case where $w$ can be written as $w(x) = \frac{1}{\epsilon^2} v(\frac{x}{\epsilon})$ with $v \in C_0^\infty(\mathbb{R},\mathbb{R}^+), v(x)=v(-x)$ and $\int_\mathbb{R} v(x) = 1$. Is there an estimation like $|w|_{1,1}\leq c\epsilon$ ?
Thanks!