The property (P) indeed characterizes the Sobolev space $W^{1,p}$. 

> **Theorem 1.** $f\in W^{1,p}(\mathbb{R}^n)$, $1<p\leq\infty$ if and only if $f\in L^p$ and there is $0\leq g\in L^p$ such that $$
 |f(x)-f(y)|\leq |x-y|(g(x)+g(y)) \ \ a.e. $$ Moreover the Sobolev norm
> is equivalent to $$ \Vert f\Vert_{M^{1,p}}=\Vert f\Vert_p+\inf_g \Vert
 g\Vert_p, $$ where the infimum is over all functions $g$ satisfying
> the above condition.

That was proved in the paper:

**P. Hajlasz,** Sobolev spaces on an arbitrary metric space, *Potential Analysis*, 5 (1996), 403-415.

Since the characterization does not use the notion of derivative the characterization was used to define Sobolev spaces on metric-measure spaces. By now this is a very well developed part of analysis with plenty of publications.

Regarding characterization (P') this is what I know:

> **Theorem 2.** $f$ belongs to the homogeneous Hardy-Sobolev space $\dot{H}^{1,1}(\mathbb{R}^n)$, if and only if  there is $0\leq g\in L^1$ such that $$
 |f(x)-f(y)|\leq |x-y|(g(x)+g(y)) \ \ a.e. $$ 

This result was proved in:

**P. Koskela, E. Saksman,**
Pointwise characterizations of Hardy-Sobolev functions.
*Math. Res. Lett.* 15 (2008), 727-744. 

Therefore functions in  the Hardy-Sobolev space $\dot{H}^{1,1}(\mathbb{R}^n)$ satisfy (P').
 
Some comments about relation between the condition (P') and Besov spaces are given at the end.

Moreover, the case $p=1$ is very close to a characterization of the space $W^{1,1}$.

> **Theorem 3.** $f\in W^{1,1}(\mathbb{R}^n)$ if and only if $f\in L^1(\mathbb{R}^n)$ and there is $g\in L^1(\mathbb{R}^n)$ such that $$
 |f(x)-f(y)|\leq |x-y|(M_{2|x-y|}g(x)+M_{2|x-y|}g(y)) $$  Where $M_Rg$
> is the Hardy-Littewood maximal function with supremum of averages over
> balls of radii less than $R$.

The proof is much more difficult than that of Theorem 1. Theorem 3 was proved in 

**P. Hajlasz,** A new characterization of the Sobolev space. (Dedicated to Professor Aleksander Pelczynski on the occasion of his 70th birthday.) Studia Math. 159 (2003), 263-275.

For a more elaborate theatment of results related to Theorems 1 and 3, see
also:

**P. Hajłasz,** Sobolev spaces on metric-measure spaces. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173-218, *Contemp. Math.,* 338, Amer. Math. Soc., Providence, RI, 2003.


There are many papers that study Besov and Triebel-Lizorkin spaces from the perspective of the characterization od $W^{1,p}$ given in Theorem 1, see for example:

**P. Koskela, D. Yang, Y. Zhou,** Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings. *Adv. Math.* 226 (2011),  3579–3621. 

**P. Koskela, D. Yang, Y. Zhou,** A characterization of Hajłasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions. *J. Funct. Anal.* 258 (2010), 2637-2661. 

Since $f\in L^{1,\infty}$ belongs (at least locally) to $L^q$ for all $q<1$ these results may apply to your question.