Let $f \colon \mathbb R^d \to \mathbb R$ be a measurable function. Consider the following property:

(P)there exist a negligible set $N \subset \mathbb R^d$ and function $T_f \in L^p(\mathbb R^d)$ such that $$ |f(x)-f(y)| \le |x-y| \left(T_f(x)+T_f(y)\right), \qquad \forall x,y \in \mathbb R^d \setminus N. $$

I believe that $f$ enjoys **(P)** if and only if $f \in W^{1,p}(\mathbb R^d)$ for $p>1$ (in this case, it is enough to take $T_f = M_{\vert Df \vert}$, being $M$ the Hardy-Littlewood maximal function). The difficult implication can be found e.g. in the book by Evans-Gariepy (p. 143, Theorem 3).

I am interested in the following relaxation of the property (P), namely:

(P$^\prime$)there exist a negligible set $N \subset \mathbb R^d$ and function $T_f \colon \mathbb R^d \to \mathbb R$ such that:

it holds $$ |f(x)-f(y)| \le |x-y| \left(T_f(x)+T_f(y)\right), \qquad \forall x,y \in \mathbb R^d \setminus N ; $$

for every $\varepsilon > 0$ there exists a function $S_f \in L^1(\mathbb R^{d})$ such that $$ \Vert T_f - S_f \Vert_{L^{1, \infty}(\mathbb R^d)} < \varepsilon, $$ being $L^{(1,\infty)}(\mathbb R^d)$ the Lorentz space.

In other words, we are relaxing the integrability assumption of $T_f$, by asking that, albeit not in $L^1$, it is arbitrarily close (in the Lorentz sense) to an $L^1$ function (namely the function $S_f$).

I would like to know if a characterization of functions fulfilling **(P$^\prime$)** is available. I (of course) expect such a characterization to be formulated in terms of Besov spaces.