Time continuity of the function in L1 norm i.e. $C([0,T];L^1) $

Question

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?

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

2. 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?

Answer 1

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:

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