Time continuity of the function in L1 norm i.e. $C([0,T];L^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 $$ N(f) := sup_{zinmathbb{R}}left(int_{mathbb{R}}frac{ |f(x+z) - f(x)|}{|z|} ,mathrm{d}xright) < infty $$
2. What is the best space to get $$ N_alpha(f) := sup_{zinmathbb{R}}left(int_{mathbb{R}}frac{ |f(x+z) - f(x)|}{|z|^alpha} ,mathrm{d}xright) < infty $$ 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 $alphain (0,1)$, 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 $$ N(f)<∞ iff f ∈ BV. $$