Time continuity of the function in L1 norm i.e. $C([0,T];L^1) $ Let $f\in L^1(\mathbb{R}),$
then its translation defined by $f(t,x):=f(x+t)$  belongs to $C([0,T];L^1(\mathbb{R})).$
In addition if $f\in BV(\mathbb{R}),$ then
\begin{eqnarray}
\int\limits_{\mathbb{R}} |f(x+t_1)-f(x+t_2)|dx &=& \sum\limits_{i\mathbb{Z}}\int\limits_{j=|t_1-t_2|i}^{|t_1-t_2|(i+1)}|f(x+t_1)-f(x+t_2)|dx\\
&=&\sum\limits_{i\in \mathbb{Z}}\int\limits_{0}^{|t_1-t_2|}|f(x+j|t_1-t_2|)-f(x+(j+1)|t_1-t_2|)|dx\\
&\leq& |t_1-t_2|TV(f)
\end{eqnarray}
which means if $f\in BV$ then the function $f(t,x)$ has Lipschitz time continuity..
I have the following doubts?

*

*Is $f\in BV$ a necessary condition for Lipschitz time continuity?   if not how to weaken the $BV$ condition to get the same result?


*Suppose $f$ is not a BV function, under some  conditions on $f$, is it possible to show $f(t,\cdot)$ is Holder time continuous ? if so what is that condition?
 A: Since
$$
\frac{1}{|t_1-t_2|}\int_{\mathbb{R}} |f(x+t_1) - f(x+t_2)|\,\mathrm{d}x = \int_{\mathbb{R}} \frac{|f(x+(t_1-t_2)) - f(x)|}{|t_1-t_2|}\,\mathrm{d}x
$$
a way to rephrase your question is:

*

*What is the best space to get
$$
N(f) := \sup_{z\in\mathbb{R}}\left(\int_{\mathbb{R}}\frac{ |f(x+z) - f(x)|}{|z|} \,\mathrm{d}x\right) < \infty
$$

*What is the best space to get
$$
N_\alpha(f) := \sup_{z\in\mathbb{R}}\left(\int_{\mathbb{R}}\frac{ |f(x+z) - f(x)|}{|z|^\alpha} \,\mathrm{d}x\right) < \infty
$$
and actually the last expressions is equivalent to a Besov seminorm (see e.g. Th. 2.3.6 in Bahouri, Chemin, Danchin, Fourier Analysis and Nonlinear Partial Differential
Equations): if $\alpha\in (0,1)$, then
$$
N_α(f)<∞ \iff f ∈ \dot{B}^\alpha_{1,\infty}
$$
From this and Besov embeddings, you can easily get a lot of other sufficient or necessary conditions in other families of spaces if you do not like Besov spaces (for example, a sufficient condition is $f$ is in the homogeneous Sobolev space $\dot W^{\alpha,1} = \dot{B}^\alpha_{1,1}$ since $\dot{W}^{\alpha,1}\subset \dot{B}^\alpha_{1,\infty}$).

When $α = 1$, however, as in your first question, it seems $BV$ is optimal (see Eq. (37.1) in An Introduction to Sobolev Spaces and Interpolation Spaces by L. Tartar) and one has
$$
N(f)<∞ \iff f ∈ BV.
$$
