Inequality with convolution

Question

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the real number

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= \|G_{\sigma_1}[f]-G_{\sigma_2}[g]\|_\infty$$

with a something depending on $\|f-g\|_\infty$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b\|f-g\|_\infty^k$$

with constants that can depend on $\sigma_1$ and $\sigma_2$.

Any hint would be highly appreciated, thank you!

Answer 1

$\newcommand\si\sigma$Such a bound does not exist. E.g., suppose that $f(u)=g(u)=cu^2$ for some real $c$ and all real $u$. Then your inequality becomes $$c|\si_1^2-\si_2^2|\le a,$$ which will not hold if $\si_1\ne\si_2$ and $c>a/|\si_1^2-\si_2^2|$.