$L^1$ convergence rate of convolution with gaussian

Question

Let $\phi_t(x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$ be the gaussian kernel and $f\ast g(x)=\int_{\mathbb{R}} f(x-y)g(y)dy$ denote the convolution. For $f\in L^1(\mathbb{R})$ I know we have $$ \lim_{t\to0^+}\Vert f\ast \phi_t - f \Vert_{L^1}=0.$$ Are there any results regarding the convergence rate for this case? For example if $ f$ is of a certain regularity say a characteristic function $f=1_{A}$ are there results like $ \Vert f\ast \phi_t - f \Vert_{L^1}=\mathcal{O}(\sqrt{t}) $? I'm especially interested in the cases where $f$ is a characteristic function and $f\in W^{1,1}(\Omega)$. I would also need references, in the case of a positive answer. Thanks in advance.