I was considering a maximal function on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ where $f=\phi$, the standard normal pdf. We first get $$\frac{1}{\lambda(V_a(x))}\int_{V_a(x)}f(y)\lambda(dy)=\frac{1}{2a}\int_{(x-a,x+a)}\phi(y)dy=\frac{\Phi(x+a)-\Phi(x-a)}{2a}$$ where $\Phi$ is the standard normal cdf. By maximizing wrt $a$ for fixed $x\geq 0$, we obtain that the optimal radius $\hat{a}$ satisfies $$\frac{\Phi(x+\hat{a})-\Phi(x-\hat{a})}{2\hat{a}}=\frac{\phi(x+\hat{a})+\phi(x-\hat{a})}{2}$$ Is there a way to find $\hat{a}(x)$ at least in semi-closed form (an integral, a series, etc)? I ran code to find it numerically and the results are peculiar:
For $0\leq x <1$, it seems that we have that the maximal function is equal to $\phi$ (i.e. $\hat{a}(x)\to 0^+$), while for $x\geq 1$ the optimal radius $\hat{a}(x)$ displays a particular functional behavior, and the maximal function shows a fat-tailed decay. I tried fitting a lot of known functions but none were close. Has this been studied?