On the Sobolev space of functions with derivative of logarithmic order

Two notions of"having a derivative of logarithmic order"have been studied. They come from the study of regularity of flows and renormalized solutions for the transport and continuity equation associated to weakly differentiable drifts.


Introduction
In this note we study two di erent notions of "having derivative of logarithmic order". They come up naturally in the study of regular Lagrangian ows and renormalized solutions for transport and continuity equation under the Sobolev regularity of the drift [1][2][3][4][5]. In order to better explain this point let us present a formal computation.
Let us x a vector eld b : R d → R d in the Euclidean space of dimension d and assume that b ∈ W ,p (R d ; R d ) for some p ≥ . Let us consider the following problem: In this setting the (ODE) problem has been studied, for a rst time, by DiPerna and Lions [1] and extended to the BV framework by Ambrosio in [2]. After these two pioneering works this topic has received a lot of attentions becoming a very thriving research eld. Let us now pass to a formal computation. In order to understand the regularity of the ow map X t (this question is central in the theory since every regularity of X, even a very mild one, allows compactness theorems, see [6] for a beautiful application to the compressible Navier-Stokes equation) it is natural to di erentiate the starting equation with respect to the space variable, obtaining Passing to the modulus and integrating we get log |∇X t (x)| ≤ˆt |∇b|(Xs(x)) ds.
Assuming that X t is a measure preserving map (in the classical context this happens when div b = ) and exploiting the Sobolev regularity of b we can conclude that with the quantitative estimate log (|∇Xt|) L p ≤ t ∇b L p .
Of course our computation is not rigorous, indeed, a priori the gradient of X t does not exist in the distributional sense. However, from (0.1) we learned that the reasonable regularity for a ow map associated to b ∈ W ,p (R d ; R d ) is neither the classical Sobolev regularity nor the fractional one but something of logarithmic order. The rst rigorous result in this direction has been proved in [3] where a class of functions very similar to the one we study in section 2 has been introduced.
The paper is organized as follows. In section 1 we study the class X γ,p de ned by means of the Gagliardo-type semi-norm Léger in [10] was the rst who used the seminorm (0.2) (in the case p = , γ = / and γ = written in the frequency domain, see subsection 1.1) to study regularity and mixing properties of solutions to the continuity equation. In [5] the authors of the present paper considered again the semi-norm (0.2) to prove new sharp regularity estimates.
In this rst section we prove the sharp Sobolev embedding inequality for the space X γ,p (see Theorem 1.5), an approximation result in Lusin's sense and the interpolation inequality between spaces L p , X γ,p and W s,p (compare with [8,Theorem 3.1]).
In section 2 we consider N ,p , another class of functions, de ned à la Hajlasz (see for instance [11]). We give a characterization of this space in terms of a niteness of a suitable discrete logarithmic Dirichlet energy (see (2.3)). We also study weak di erentiability properties of functions in N ,p and we eventually establish a link with the rst introduced space X γ,p .

Notations.
We denote by Br(x) the ball of radius r > centered at x ∈ R d , where d ≥ . We often write Br instead of Br( ). Let us set denotes the Hardy-Littlewood maximal function. We shall use the expression a c b to mean that there exists a universal constant C depending only on c such that a ≤ Cb. The same convention is adopted for c and c.

Acknowledgements:
The authors are grateful to Giuseppe Mingione and Jean Van Schaftingen for valuable suggestions on the paper.

The space X γ,p
Let us de ne the rst space.
De nition 1.1. Let p ∈ ( , ∞) and γ ∈ ( , ∞) be xed. We de ne It is immediate to verify that X γ,p endowed with Therefore, the semi-norm (1.1) is not trivial only when γ ≥ .
Let us brie y discuss the analogies between X γ,p and the Sobolev spaces of fractional order W s,p (see [12] for a reference on this topic). Let p > and s ∈ ( , ) be xed, the space W s,p consists of functions f ∈ L p such that and it is endowed with f p W s,p := f p L p + [f ] p W p,s . Understanding X γ,p as the space of functions in L p with derivative of logarithmic order γ in L p it is natural to expect the continuous inclusions W s,p ⊂ X γ ,p ⊂ X γ,p and s ∈ ( , ). This is, indeed, the statement of the following proposition whose proof is a simple exercise. Proposition 1.3. Let p ∈ ( , ∞) be xed. For any ≤ γ ≤ γ and s ∈ ( , ) it holds

. The case p =
In this section we characterize the space X γ, by means of the Fourier transform.
Proof. Using Plancherel's formula we get, It is enough to show that In order to prove (1.5) we use the elementary inequality − cos(a) a for any a ∈ (− , ) obtaininĝ Let us now pass to the proof of (1.6). First observe that where we used the elementary inequality − cos(a) a for any a ∈ R and the assumption |ξ | > .
Let us now show the converse inequality. Using the Coarea formula we can writê where H d− denotes the (d − )-dimensional Hausdor measure. It is elementary to see that, for every ξ ∈ R d and r > satisfying r|ξ | ≥ , it holdŝ This gives (1.6). The proof is complete.

. Sobolev embedding and interpolation inequality
The rst result of this section is a Sobolev embedding theorem for X γ,p : In particular the following log-Sobolev inequality holds true for any f ∈ X γ,p , Let us point out that the inequality (1.8) can be obtained as a consequence of the embedding theorems studied in [7]. Nevertheless, our strategy is completely di erent and based on (1.7) that, as far as we know, is a new result. It is also worth mentioning that our proof of Theorem 1.5 is completely elementary and short.
Proof. We may assume without loss of generality that f L p = .
Clearly it is enough to show where Cp,γ ≥ is a xed constant depending only on p and γ.
For any t ∈ ( , / ), using the assumption f L p = we get Here the constant does not depend on d since Let us now x < λ < / . We integrate the inequality above against t log( /t) −pγ with respect to t ∈ (λ, / ) obtaining so, rearranging the terms, we end up with (1.9) immediately follows. The proof is complete.
The just explained strategy can be used also to obtain a very short proof of the fractional Sobolev embedding theorem. In the fractional context, a very similar argument has already appeared in the literature (see [13, pag. 241]). Proposition 1.6. Let us x s ∈ ( , ) and p ∈ ( , d/s). We set p * := dp d−sp . For any f ∈ W s,p (R d ) the following point-wise inequality holds true In particular we deduce the well-known Sobolev inequality Proof. For any x ∈ R d and t > we can write By mean of Hölder's inequality we get and we end up with Some technical remarks are in order.
Remark 1.7. The estimate (1.7) could be improved in the following way: for any γ > , p > and every f ∈ for L d -a.e. every x ∈ R d . Let us explain how to modify the proof of (1.7) to get (1.12). We rst assume p > and we use the inequalitŷ The proof of (1.12) is achieved arguing exactly as in the proof of (1.7). We nally extend (1.12) to every p ∈ ( , ] by means of the elementary inequality Observe that The function Br( ) is a very natural candidate to show the sharpness of (1.8), since, in general, Sobolev embeddings are related to the isoperimetric problem and balls are minimizers, at least in the classical context. However, it is worth mentioning that a notion of logarithmic perimeter of order γ > associated to a set can be obtained writing P L γ (E) := [ E ] X γ, . We expect that balls are the only minimizers of P L γ as it happens in the classical and fractional case (see for instance [15]), but we do not investigate this problem here.
The last result we present in this section is an interpolation inequality between spaces L p (R d ), X γ,p and W s,p . Even though this result can be deduced from [8, Theorem 3.1] we prefer to present a very simple and direct proof. Proposition 1.9. Let p > , s ∈ ( , ) and γ > be xed. For any f ∈ L p (R d ) we have (1.14) Proof. Assume without loss of generality f L p = . Let λ ∈ ( , / ) be xed, we have

. Lusin's estimate
It is well-known that a quantitative Lusin's approximation property (see [16], [18]) characterizes Sobolev spaces, even in the very abstract setting of measure metric spaces (see for instance [19]). In this section we study this approximation property for X γ,p functions. For any f ∈ X γ,p we de ne The main result of the section is the following.
Theorem 1.11. Let p > and γ > be xed. For any f ∈ X γ,p it holds for any x, y ∈ R d such that |x − y| < .
Let us begin with a simple lemma.

Lemma 1.12. Let p > , x, y ∈ R d be xed. For any f ∈ L p it holds
for any r ≥ |x − y|.
Proof. Let us estimate the proof is complete.
We are now ready to prove Theorem 1.11.
Proof of Theorem 1.11. We integrate both sides of (1.16) with respect to r ∈ ( / , |x − y|) against r log( /r) −pγ getting where Proof. As in the proof of Theorem 1.11, we integrate both sides of (1.16) with respect to r against a suitable kernel K(r), in this case we should consider K(r) = r> |x−y| r d+ps . Let us nally prove a partial converse of Theorem 1.11. Proposition 1.14. Let p > and γ > be xed.
for some g ∈ L p (R d ). Then f ∈ X α,p for any α < γ with estimate Proof. Let us x < α < γ, we estimate changing variables according to λ log( /|h|) γ = t we get The proof is complete.

The space N s,p
The aim of this section is to study another class of functions with derivative of logarithmic order. It comes up naturally in the study of regularity for Lagrangian ows associated to Sobolev vector elds, see [3].

De nition 2.1.
Let p ≥ and s ∈ ( , ] be xed. We say that f ∈ L p loc (R d ) belongs to N s,p if there exists a positive function g ∈ L p (R d ) such that We set [f ] N p := inf{ g L p }, where the in mum runs over all possible g satisfying (2.1).
[ · ] N ,p in general is not a semi-norm. It satis es the triangle inequality and as it can be seen using the elementary identity e t − = n≥ t n n! , for any t ∈ R. But in general in (2.2) the inequality is strict.
For any f ∈ L p loc (R d ) let us de ne the functional Roughly speaking it can be seen as a discrete fractional logarithmic Dirichlet's energy. The aim of the next proposition is to link the condition (2.1) to integrability properties of the function Φ * s f . thanks to the boundedness of M in L p (see [20]). In order to achieve the proof of (2.4) it remains to show the converse of (2.5).
For any x, y ∈ R d let us set r = |x − y|, we get It is immediate to see that Φs,q f L p/q d,p g q L p , when ≤ q < p, and Φs,p f L ,∞ d,p g p L p . Let us now recall the de nition of weak di erentiability.
De nition 2.5. We say that a measurable function f : for any < R < ∞. We set ∇f (x) · y := L(y).
It is well-known that a function f is weakly di erentiable at L d -a.e. x ∈ R d if and only if it can be approximated with Lipschitz functions in Lusin's sense. Namely, for any ε > there exists a Lipschitz function g : [21] for a good reference on this topic. The aim of our next proposition is to study, in a quantitative manner, weakly di erentiability properties of functions f ∈ N ,p . Precisely we have the following.
Proof. It is straightforward to see that f is weakly di erentiable (see discussion below). For any constant M > we have − it exists thanks to the Dunford-Pettis lemma, see [18]. We can also assume that t → Ψ(t) t is increasing. Setting and using the very de nition of N ,p we get thus it is nite for L d -a.e. x ∈ R d thanks to the (1,1) weak estimate for the maximal function. Therefore, it is enough to prove that (ii) holds for every x ∈ R d such that Let us x a parameter < λ < / , using the Jensen inequality we get Taking rst the limit for r → and after λ → we get the sought conclusion since fr → locally in measure.
Remark 2.7. It is natural to wonder if the statement of Proposition 2.6 has a converse, or if some quantitative version of the weakly di erentiability at L d -a.e. almost every point could guarantee the property (2.1). The answer is negative, indeed for any p ∈ [ , ∞) it is possible to build a function f ∈ L p (R d ) that is weakly di erentiable almost everywhere with ∇f = but does not belong to N ,q for any q.
Let us illustrate how to build such example. Let us x an integer M > . It is enough to built a function f supported in [ , ] that is weakly di erentiable with f = L -a.e., f L ∞ = and The last result of this paper concerns with the link between the spaces X γ,p and N s,p . A version of this result was crucial in our paper [5] to study a sharp regularity for the continuity equation associated to a divergencefree Sobolev drift.