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: 
> 
> 
> 1. it holds $$
> |f(x)-f(y)| \le |x-y| \left(T\_f(x)+T\_f(y)\right), \qquad \forall x,y \in \mathbb R^d \setminus N ; 
> $$
> 2. 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](https://en.wikipedia.org/wiki/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.



Estimate of the difference quotients in terms of an $L^{1,\infty}$ function
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. 


