Does there exist such a mollifier function?

Question

I want to know if there exists a (mollifier) function $\psi \in C^{\infty}(\mathbb{R})$ such that

i) $\operatorname{Supp}\psi \subset B_1(0)$

ii) $\int_{\mathbb{R}} \psi(x)dx=1 $

and

iii) $\int_{\mathbb{R}} |\psi'(x)|dx < \frac12 $

Answer 1

Suppose $\psi$ satisfied properties i) and iii) listed in the question. Then $\psi(x) = 0$ for $|x|\ge 1$, and for $|x|<1$ we have $$\psi(x) = \psi(x) - \psi(-1) = \int\limits_{-1}^{x}{\psi'(t)\text{ d}t} $$ and hence $$|\psi(x)|\le\int\limits_{-1}^{x}{|\psi'(t)|\text{ d}t}\le\int\limits_{\mathbb{R}}{|\psi'(t)|\text{ d}t}<\frac{1}{2}. $$ It follows that $$\int\limits_{\mathbb{R}}{\psi(x)\text{ d}x}\le\int\limits_{\mathbb{R}}{|\psi(x)|\text{ d}x} = \int\limits_{-1}^{1}{|\psi(x)|\text{ d}x} < \int\limits_{-1}^{1}{\frac{1}{2}\text{ d}x} = 1. $$ Note that the last inequality is strict because $\psi$ is continuous. It follows that any $\psi\in C^{\infty}(\mathbb{R})$ satisfying i) and iii) cannot also satisfy ii).