A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

We continue the study of the space $BV^\alpha(\mathbb R^n)$ of functions with bounded fractional variation in $\mathbb R^n$ and of the distributional fractional Sobolev space $S^{\alpha,p}(\mathbb R^n)$, with $p\in [1,+\infty]$ and $\alpha\in(0,1)$, considered in the previous works arXiv:1809.08575 and arXiv:1910.13419. We first define the space $BV^0(\mathbb R^n)$ and establish the identifications $BV^0(\mathbb R^n)=H^1(\mathbb R^n)$ and $S^{\alpha,p}(\mathbb R^n)=L^{\alpha,p}(\mathbb R^n)$, where $H^1(\mathbb R^n)$ and $L^{\alpha,p}(\mathbb R^n)$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla^\alpha$ strongly converges to the Riesz transform as $\alpha\to0^+$ for $H^1\cap W^{\alpha,1}$ and $S^{\alpha,p}$ functions. We also study the convergence of the $L^1$-norm of the $\alpha$-rescaled fractional gradient of $W^{\alpha,1}$ functions. To achieve the strong limiting behavior of $\nabla^\alpha$ as $\alpha\to0^+$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

The asymptotic behavior of the fractional gradient ∇ α as α → 1 − was fully discussed in [28] (see also [14,Theorem 3.2] for a different proof of (1.10) below for the case p ∈ (1, +∞) via Fourier transform). Precisely, if f ∈ W 1,p (R n ) for some p ∈ [1, +∞), then f ∈ S α,p (R n ) for all α ∈ (0, 1) with lim α→1 − ∇ α f − ∇f L p (R n ; R n ) = 0. (1.10) If f ∈ BV (R n ) instead, then f ∈ BV α (R n ) for all α ∈ (0, 1) with D α f ⇀ Df in M (R n ; R n ) and |D α f | ⇀ |Df | in M (R n ) as α → 1 − and lim α→1 − |D α f |(R n ) = |Df |(R n ). (1.11) We underline that, differently from the limits (1.6) and (1.8), the renormalizing factor (1 − α) 1 p does not appear in (1.10) and (1.11). This is motivated by the fact that the constant µ n,α encoded in the definition (1.3) of the operator ∇ α satisfies Concerning the asymptotic behavior of ∇ α as α → 0 + , at least for sufficiently regular functions, the fractional gradient in (1.3) is converging to the operator |y − x| n+1 dy, x ∈ R n . (1.12) Here and in the following, µ n,0 is simply the limit of the constant µ n,α defined in (1.5) as α → 0 + (thus, in this case, no renormalization factor has to be taken into account). The operator in (1.12) is well defined (in the principal value sense) at least for all f ∈ C ∞ c (R n ) and, actually, coincides (possibly up to a minus sign, see Section 2.1 below) with the well-known vector-valued Riesz transform Rf , see [40,73,74]. The formal limit ∇ α → R as α → 0 + can be also motivated either by the asymptotic behavior of the Fourier transform of ∇ α as α → 0 + or by the fact that stands for the Riesz potential of order α ∈ (0, n). In a similar fashion, the fractional α-divergence in (1.4) is converging as α → 0 + to the operator which is well defined (in the principal value sense) at least for all ϕ ∈ C ∞ c (R n ; R n ). As a natural target space for the study of the limiting behavior of ∇ α as α → 0 + , in analogy with the fractional variation (1.1), we introduce the space BV 0 (R n ) of functions f ∈ L 1 (R n ) such that the quantity Surprisingly, it turns out that D 0 f ≪ L n for all f ∈ BV 0 (R n ), in contrast with what is known for the fractional α-variation in the case α ∈ (0, 1], see [27,Theorem 3.30]. More precisely, we prove that where is the well-known (real) Hardy space.
Having the identification (1.13) at disposal, we can rigorously establish the validity of the convergence ∇ α → R as α → 0 + . For p = 1, we prove that For p ∈ (1, +∞) instead, since the Riesz transform (1.12) extends to a linear continuous operator R : L p (R n ) → L p (R n ; R n ), the natural target space for the study of the limiting behavior of the fractional gradient is simply L p (R n ; R n ). In this case, we prove that The limits in (1.14) and (1.15) can be considered as the counterparts of (1.7) in our fractional setting. However, differently from (1.7), in (1.14) and in (1.15) we obtain strong convergence. This improvement can be interpreted as a natural consequence of the fact that, generally speaking, the L p -norm of the fractional gradient ∇ α allows for more cancellations than the W α,p -seminorm.
Since the Riesz transform (1.12) extends to a linear continuous operator R : H 1 (R n ) → H 1 (R n ; R n ), the limit in (1.14) can be improved. Precisely, we prove that Here is (an equivalent definition of) the fractional Hardy-Sobolev space, see [75] and below for a more detailed presentation. One can recognize that so that (1.16) is indeed a reinforcement of (1.14).
Naturally, if f / ∈ H 1 (R n ), then we cannot expect that ∇ α f → Rf in L 1 (R n ; R n ) as α → 0 + . Instead, as suggested by the limit in (1.7), we have to consider the asymptotic behavior of the rescaled fractional gradient α ∇ α f as α → 0 + . In this case, we prove that ( 1.17) for all f ∈ α∈(0,1) W α,1 (R n ). Note that (1.17) is consistent with both (1.7) and (1.14). Indeed, on the one side, by simply bringing the modulus inside the integral in the definition (1.3) of ∇ α , we can estimate for all f ∈ W α,1 (R n ) (see also [27,Theorem 3.18]), so that, by (1.7), we can infer lim sup [74, Chapter III, Section 5.4(c)] for example), and thus for all f ∈ H 1 (R n ) ∩ α∈(0,1) W α,1 (R n ) the limit in (1.17) reduces to in accordance with the strong convergence (1.14). Let α ∈ (0, 1) be fixed. In the standard fractional framework, by a simple splitting argument, it is not difficult to estimate the W β,1 -seminorm of a function f ∈ W α,1 (R n ) as for all β ∈ (0, α). Inequality (1.19) implies the bound 20) in agreement with (1.7). In a similar fashion (but with a more delicate analysis), an interpolation inequality of the form (1.19) has been recently obtained by the third and the fourth author for the for all β ∈ (0, α), where c n,α,β > 0 is a constant such that see [28,Proposition 3.12] (see [28,Proposition 3.2] also for the case α = 1). Here and in the following, we let [f ] BV α (R n ) be the total fractional variation (1.1) of f ∈ BV α (R n ). Thanks to (1.23), inequality (1.21) implies the bound coherently with (1.17).
Although strong enough to settle the asymptotic behavior of the fractional gradient ∇ β when β → α − thanks to (1.22), because of (1.24) inequality (1.21) is of no use for the study of the strong L 1 -limit ∇ β → R as β → 0 + . To achieve this convergence, we thus have to control the interpolation constant c n,α,β in (1.21) with a new interpolation constant c n,α > 0 independent of β ∈ (0, α), at the price of weakening (1.21) by replacing the L 1 -norm with a bigger norm.
This strategy is in fact motivated by the non-optimality of the bound (1.24) since, in view of the limit in (1.17), we can still expect some cancellation effect of the fractional gradient for a subclass of L 1 -functions having zero average. Note that this approach cannot be implemented to stabilize the standard interpolation inequality (1.19), since the bound in (1.20) is in fact optimal due to (1.7).
At this point, our idea is to exploit the cancellation properties of the fractional gradient ∇ β by rewriting its non-local part in terms of a convolution kernel. In more precise terms, recalling the definition in (1.3), for R > 0 we can split for all Schwartz functions f ∈ S(R n ), where the convolution kernel K β,R is a smoothing of the function y → y |y| n+β+1 χ [R,+∞) (|y|). By the Calderón-Zygmund Theorem, we can extend the functional defined in (1.26) to a linear continuous mapping ∇ β ≥R : H 1 (R n ) → L 1 (R n ; R n ) whose operator norm can be estimated as 27) for some dimensional constant c n > 0. By combining the splitting (1.25) with the bound (1.27) and arguing as in [28], we get that for all β ∈ [0, α) and all f ∈ H 1 (R n ) ∩ BV α (R n ), whenever α ∈ (0, 1]. Exploiting (1.28) together with an approximation argument, we thus just need to establish (1.14) for all sufficiently regular functions, in which case we can easily conclude by a direct computation. To achieve the limit in (1.15) for p ∈ (1, +∞) and the stronger convergence in (1.16) for the case p = 1, we adopt a slightly different strategy. Instead of splitting the fractional gradient as in (1.25), we rewrite it as is the usual fractional Laplacian with renormalizing constant given by Since the Riesz transform extends to a linear continuous operator on L p (R n ) and H 1 (R n ) as mentioned above, to achieve (1.15) and (1.16) we just have to study the continuity properties of (−∆) Exploiting the good decay properties of the derivatives of m α,β (uniform with respect to the parameters 0 ≤ β ≤ α ≤ 1), by the Mihlin-Hörmander Multiplier Theorem the convolution operator in (1.31) can be extended to two linear operators continuous from L p (R n ) to itself and from H 1 (R n ) to itself, respectively. Going back to (1.29) and (1.30), we can exploit the continuity properties of the (extensions of) the operator T m α,β to deduce two new interpolation inequalities. On the one hand, given p ∈ (1, +∞), there exists a constant c n,p > 0 such that for all 0 ≤ γ ≤ β ≤ α ≤ 1 and all f ∈ S α,p (R n ). In the particular case γ = 0, thanks to the L p -continuity of the Riesz transform, we also have for all 0 ≤ β ≤ α ≤ 1 and all f ∈ S α,p (R n ). On the other hand, there exists a dimensional constant c n > 0 such that for all 0 ≤ γ ≤ β ≤ α ≤ 1 and all f ∈ HS α,1 (R n ). Again, in the particular case γ = 0, thanks to the H 1 -continuity of the Riesz transform, we also have for all 0 ≤ β ≤ α ≤ 1 and all f ∈ HS α,1 (R n ). Having the interpolation inequalities (1.33) and (1.35) at disposal, as before we just need to establish (1.15) and (1.16) for all sufficiently regular functions, in which case we can again conclude by a direct computation.
As the reader may have noticed, in the above line of reasoning we can infer the validity of (1.32) and (1.34) only if we are able to prove the identifications (1.36) for p ∈ (1, +∞), and (1.37) respectively, with equivalence of the naturally associated norms, where (Id − ∆) − α 2 is the standard Bessel potential. While (1.37) follows by a plain approximation argument building upon the results of [75], the identification in (1.36) is more delicate and, actually, answers an equivalent question left open in [27], that is, the density of C ∞ c (R n ) functions in S α,p (R n ), see Appendix A for the proof. In other words, the equivalence (1.36) allows to identify the Bessel potential space with the distributional fractional Sobolev space S α,p (R n ) in (1.2). Thanks to the identification L α,p (R n ) = S α,p (R n ), many of the results established in [13,14] and in [67,68] can be proved in a simpler and more direct way. See also Appendix B for other consequences of this identification.

Complex interpolation and open problems.
To achieve the interpolation inequalities (1.28) and (1.32) -(1.35), we essentially relied on a direct approach exploiting the precise structure of the fractional gradient in (1.3). Adopting the point of view of [52,62], a possible alternative route to the above fractional inequalities may follow from complex interpolation techniques. According to [15,Theorem 6.4.5(7)] and thanks to the aforementioned identification L α,p (R n ) = S α,p (R n ), for all α, ϑ ∈ (0, 1) and p ∈ (1, +∞) we have the complex interpolation Here and in the following, we write A ∼ = B to emphasize the fact that the spaces A and B are the same with equivalence (and thus, possibly, not equality) of the relative norms. As a consequence, (1.38) implies that, for all 0 < β < α < 1 and p ∈ (1, +∞), there exists a constant c n,α,β,p > 0 such that for all f ∈ S α,p (R n ). In a similar way (we omit the proof because beyond the scopes of the present paper), for all α, ϑ ∈ (0, 1) one can also establish the complex interpolation and thus, for some constant c n,α,β > 0, for all f ∈ HS α,1 (R n ). Inequalities (1.39) and (1.41) suggest that, in order to obtain (1.33) and (1.35) with complex interpolation methods, one essentially should prove that the identifications (1.38) and (1.40) hold uniformly with respect to the interpolating parameter. We believe that this result may be achieved but, since we do not need this level of generality for our aims, we preferred to prove (1.32) -(1.35) in a more direct and explicit way.
1.5. Organization of the paper. We conclude this introduction by briefly presenting the organization of the present paper. Section 2 provides the main notation, recalls the needed properties of the fractional operators ∇ α and div α and, finally, deals with the properties of the space HS α,1 (R n ). Section 3 is devoted to the proof of the identification BV 0 (R n ) = H 1 (R n ), together with some useful consequences about the relation between H 1 (R n ) and W α,1 (R n ). In Sections 4 and 5, the core of our work, we detail the proof of the interpolation inequalities (1.28), (1.32) and (1.34) and, consequently, we prove both the strong convergence of the fractional gradient ∇ α as α → 0 + given by (1.15), (1.16) and the limit (1.17). We close our work with three appendices: in Appendix A we prove the density of C ∞ c (R n ) functions in S α,p (R n ); in Appendix B we state some properties of S α,p -functions; in Appendix C we establish some continuity properties of the map α → ∇ α .

Preliminaries
We start with a brief description of the main notation used in this paper. In order to keep the exposition as reader-friendly as possible, we retain the same notation adopted in the previous works [27,28].
2.1. General notation. We let L n and H α be the n-dimensional Lebesgue measure and the α-dimensional Hausdorff measure on R n respectively, with α ≥ 0. A measurable set is a L n -measurable set. We also use the notation |E| = L n (E). All functions we consider in this paper are Lebesgue measurable. We let B r (x) be the standard open Euclidean ball with center x ∈ R n and radius r > 0. We set B r = B r (0). Recall that ω n := |B 1 | = π n 2 /Γ n+2 2 and H n−1 (∂B 1 ) = nω n , where Γ is the Euler's Gamma function, see [9].
For m ∈ N, the total variation on Ω of the m-vector-valued Radon measure µ is defined as We thus let M (Ω; R m ) be the space of m-vector-valued Radon measure with finite total variation on Ω.
For k ∈ N 0 ∪ {+∞} and m ∈ N, we let C k c (Ω; R m ) and Lip c (Ω; R m ) be the spaces of C k -regular and, respectively, Lipschitz-regular, m-vector-valued functions defined on R n with compact support in the open set Ω ⊂ R n .
For m ∈ N, we let S(R n ; R m ) be the space of m-vector-valued Schwartz functions on R n .
where x a := x a 1 1 · . . . · x an n for all multi-indices a ∈ N n 0 . We let S ′ (R n ; R m ) be the dual of S(R n ; R m ) and we call it the space of tempered distributions. See [40, Section 2.2 and 2.3] for instance.
For any exponent p ∈ [1, +∞], we let L p (Ω; R m ) be the space of m-vector-valued Lebesgue p-integrable functions on Ω.
We let be the Fourier transform of the function f ∈ L 1 (R n ; R m ). As it is well known, the Fourier transform maps S(R n ; R m ) onto itself and may be extended to S ′ (R n ; R m ) (see [40, Sections 2.2 and 2.3] for instance). We let be the space of m-vector-valued Sobolev functions on Ω, see for instance [46,Chapter 10] for its precise definition and main properties. We let be the space of m-vector-valued functions of bounded variation on Ω, see for instance [4,Chapter 3] or [34,Chapter 5] for its precise definition and main properties. For α ∈ (0, 1) and p ∈ [1, +∞), we let be the space of m-vector-valued fractional Sobolev functions on Ω, see [32] for its precise definition and main properties. For α ∈ (0, 1) and p = +∞, we simply let , the space of m-vector-valued bounded α-Hölder continuous functions on Ω.
Given α ∈ (0, n), let We recall that, if α, β ∈ (0, n) satisfy α + β < n, then we have the following semigroup property then there exists a constant C n,α,p > 0 such that the operator in (2.1) satisfies . As a consequence, the operator in (2.1) extends to a linear continuous operator from L p (R n ; R m ) to L q (R n ; R m ), for which we retain the same notation. For a proof of (2.2) and (2.3), we refer the reader to [ Given α ∈ (0, 1), we also let For α ∈ (0, 1) and p ∈ (1, +∞), let see [63,Theorem 27.3]. In particular, the function defines a norm on L α,p (R n ; R m ) equivalent to the one in (2.6) (and so, unless otherwise stated, we will use both norms (2.6) and (2.8) with no particular distinction).
We recall that C ∞ c (R n ) is a dense subset of L α,p (R n ; R m ), see [1, Theorem 7.63(a)] and [63,Lemma 27.2]. Note that the space L α,p (R n ; R m ) can be defined also for any α ≥ 1 by simply using the composition properties of the Bessel potential (or of the fractional Laplacian), see [1,Section 7.62]. All the properties stated above remain true also for α ≥ 1 and, moreover, For m ∈ N, we let be the m-vector-valued (real) Hardy space endowed with the norm We refer the reader to [ Chapter III] for a more detailed exposition. We warn the reader that the definition in (2.9) agrees with the one in [74] and differs from the one in [41,73] for a minus sign. We also recall that the Riesz transform (2.9) defines a continuous operator R : L p (R n ; R m ) → L p (R n ; R mn ) for any given p ∈ (1, +∞), see [40,Corollary 5.2.8], and a continuous operator In the sequel, in order to avoid heavy notation, if the elements of a function space F (Ω; R m ) are real-valued (i.e. m = 1), then we will drop the target space and simply write F (Ω).

2.2.
Overview of ∇ α and div α and the related function spaces. We recall the definition (and the main properties) of the non-local operators ∇ α and div α , see [27,28,70] and the monograph [61, Section 15.2].
Let α ∈ (0, 1) and set We let The non-local operators ∇ α and div α are well defined in the sense that the involved integrals converge and the limits exist. Moreover, since Thanks to [27, Proposition 2.2], given α ∈ (0, 1) we can equivalently write for all f ∈ Lip c (R n ; R n ) and ϕ ∈ Lip c (R n ; R n ), respectively. The fractional operators ∇ α and div α are dual in the sense that for all f ∈ Lip c (R n ) and ϕ ∈ Lip c (R n ; R n ), see [69,Section 6] and [27,Lemma 2.5]. In addition, given f ∈ Lip c (R n ) and ϕ ∈ Lip c (R n ; R n ), we have for all p ∈ [1, +∞], see [27,Corollary 2.3]. The above results and identities hold also for functions f ∈ S(R n ) and ϕ ∈ S(R n ; R n ). Given α ∈ (0, 1) and p ∈ [1, +∞], inspired by the integration-by-parts formula (2.11), we say that a function f ∈ L p (R n ) has bounded fractional α-variation if is a Banach space and that the fractional variation defined in (2.13) is lower semicontinuous with respect to L p -convergence. In the sequel, we also use the notation In the case p = 1, we simply write BV α,1 (R n ) = BV α (R n ). The space BV α (R n ) resembles the classical space BV (R n ) from many points of view and we refer the reader to [27, Section 3] for a detailed exposition of its main properties.
Again motivated by (2.11) and in analogy with the classical case, given α ∈ (0, 1) and p ∈ [1, +∞] we define the weak fractional α-gradient of a function f ∈ L p (R n ) as the We notice that, in the case f ∈ Lip c (R n ) (or f ∈ S(R n )), the weak fractional α-gradient of f coincides with the one defined above, thanks to (2.11). As above, the reader can verify that the distributional fractional Sobolev space endowed with the norm is a Banach space. In the case p = 1, starting from the very definition of the fractional gradient ∇ α , one can check that  [27,Theorem 3.23].
In the case p ∈ (1, +∞), the density of the set of test functions in the space S α,p (R n ) was left as an open problem in [27,Section 3.9]. More precisely, defining endowed with the norm in (2.15), it is immediate to see that S α,p 0 (R n ) ⊂ S α,p (R n ) with continuous embedding. The space (S α,p 0 (R n ), · S α,p (R n ) ) was introduced in [67] (with a different, but equivalent, norm) and, in fact, it satisfies S α,p 0 (R n ) = L α,p (R n ) for all α ∈ (0, 1) and p ∈ (1, +∞), see [67,Theorem 1.7]. In Theorem A.1 in the appendix, we positively solve the problem of the density of C ∞ c (R n ) in the space S α,p (R n ). As a consequence, we obtain the following result.
According to Corollary 2.1, in the sequel we will also use the symbol S α,p to denote the Bessel potential space L α,p . In addition, consistently with the asymptotic behavior of the fractional gradient ∇ α as α → 1 − established in [28], we will sometimes denote the Sobolev space W 1,p as S 1,p for p ∈ [1, +∞).
Thanks to the identification given by Corollary 2.1, we can prove the following result.
Proof. By Corollary 2.1, we equivalently have to prove that the set S 0 (R n ) is dense in L α,p (R n ). To this aim, let us consider the functional M : Clearly, the linear functional M cannot be continuous and thus its kernel S 0 (R n ) must be dense in S(R n ) with respect to the L p -norm. Since the Bessel potential (Id − ∆) − α 2 : (S(R n ), · S α,p (R n ) ) → (S(R n ), · L p (R n ) ) is an isomorphism, the conclusion follows.
2.3. The fractional Hardy-Sobolev space HS α,1 (R n ). Following the classical approach of [75], for α ∈ [0, 1] let be the (real) fractional Hardy-Sobolev space endowed with the norm In particular, HS 0,1 (R n ) = H 1 (R n ) coincides with the (real) Hardy space and H 1,1 (R n ) is the standard (real) Hardy-Sobolev space. As remarked in [75, p. 130], HS α,1 (R n ) can be equivalently defined as In particular, the function For the reader's convenience we briefly prove the following density result.
Proof. Since the set S ∞ (R n ) is dense in H 1 (R n ) by [74, Chapter III, Section 5.2(a)], the set is dense (and embeds continuously) in HS α,1 (R n ). Thus the conclusion follows.
Exploiting Lemma 2.3, for α ∈ (0, 1), the space HS α,1 (R n ) can be equivalently defined as the space Indeed, if f ∈ S ∞ (R n ), then, by exploiting Fourier transform techniques, we can write for all f ∈ S ∞ (R n ), thanks to the H 1 -continuity property of the Riesz transform and the fact that where R j is the j-th component of the Riesz transform R. By Lemma 2.3, the validity of (2.18) extends to all f ∈ HS α,1 (R n ) and the conclusion follows. As a consequence, note that HS α,1 (R n ) ⊂ S α,1 (R n ) for all α ∈ (0, 1) with continuous embedding. We note that the well-posedness and the equivalence of the definitions of HS α,1 (R n ) given above and the stated results hold for any α ≥ 0 thanks to the composition properties of the operators involved. We leave the standard verifications to the interested reader.
3. The BV 0 (R n ) space 3.1. Definition of BV 0 (R n ) and Structure Theorem. Somehow naturally extending the definitions given in (2.10) to the case α = 0, for f ∈ Lip c (R n ) and ϕ ∈ Lip c (R n ; R n ) we define ∇ 0 f := I 1 ∇f and div 0 ϕ := I 1 divϕ.
It is immediate to check that the integration-by-parts formula holds for all given f ∈ Lip c (R n ) and ϕ ∈ Lip c (R n ; R n ). Hence, in analogy with [27, Definition 3.1], we are led to the following definition (which is well posed, since div 0 ϕ ∈ L ∞ (R n ) for ϕ ∈ Lip c (R n ; R n )).
The proof of the following result is very similar to the one of [27, Theorem 3.2] and is omitted.
for all ϕ ∈ C ∞ c (R n ; R n ). In addition, for all open sets U ⊂ R n it holds As already announced in [28], the space BV 0 (R n ) actually coincides with the Hardy space H 1 (R n ). More precisely, we have the following result.

Theorem 3.3 (The identification BV
for every f ∈ BV 0 (R n ).
Proof. We prove the two inclusions separately.
Proof of H 1 (R n ) ⊂ BV 0 (R n ). Let f ∈ H 1 (R n ) and assume f ∈ Lip c (R n ). By (3.1), we immediately get that D 0 f = Rf L n in M (R n ; R n ) with Rf = ∇ 0 f in L 1 (R n ; R n ), so that f ∈ BV 0 (R n ). Now let f ∈ H 1 (R n ). By [74, Chapter III, Section 5.2(b)], we can for all k ∈ N. Passing to the limit as k → +∞, we get , Rf is well defined as a (vector-valued) distribution, see [74, Chapter III, Section 4.3]. Thanks to (3.2), we also have that Rf, ϕ = D 0 f, ϕ for all ϕ ∈ C ∞ c (R n ; R n ), so that Rf = D 0 f in the sense of distributions. Now let (̺ ε ) ε>0 ⊂ C ∞ c (R n ) be a family of standard mollifiers (see e.g. [27, Section 3.2]). We can thus estimate for all ε > 0, so that f ∈ H 1 (R n ) by [74, Chapter III, Section 4.3, Proposition 3], with D 0 f = Rf L n in M (R n ; R n ).  ∈ (0, 1). The following hold.

Relation between
( Proof. We prove the two statements separately. Proof of (i). Let f ∈ H 1 (R n ). By the Stein-Weiss inequality (see [66, Theorem 2] for instance), we know that u := I α f ∈ L n n−α (R n ). To prove that |D α u|(R n ) < +∞, we exploit Theorem 3.3 and argue as in the proof of [27,Lemma 3.28]. Indeed, for all ϕ ∈ C ∞ c (R n ; R n ), we can write by Fubini's Theorem, since f ∈ L 1 (R n ) and I α |div α ϕ| ∈ L ∞ (R n ), being thanks to the semigroup property (2.2) of the Riesz potentials. This proves that D α u = D 0 f = Rf L n in M (R n ; R n ), again thanks to Theorem 3.3.
We end this section with the following consequence of Proposition 3.4.

Interpolation inequalities
4.1. The case p = 1 via the Calderón-Zygmund Theorem. Here and in the rest of the paper, let (η R ) R>0 ⊂ C ∞ c (R n ) be a family of cut-off functions defined as η R (x) = η |x| R , for all x ∈ R n and R > 0, For α ∈ (0, 1) and R > 0, let T α,R : S(R n ) → S ′ (R n ; R n ) be the linear operator defined by for all f ∈ S(R n ). In the following result, we prove that T α,R is a Calderón-Zygmund operator mapping H 1 (R n ) to L 1 (R n ; R n ).

Lemma 4.1 (Calderón-Zygmund estimate for T α,R ).
There is a dimensional constant τ n > 0 such that, for any given α ∈ (0, 1) and R > 0, the operator in (4.3) uniquely extends to a bounded linear operator T α,R : Proof. We apply [41, Theorem 2.4.1] to the kernel First of all, we have so that we can choose A 1 = 2nω n R −α in the size estimate (2.4.1) in [41]. We also have where c n > 0 is some dimensional constant, so that we can choose A 2 = c ′ n R −α in the smoothness condition (2.4.2) in [41], where c ′ n > c n is another dimensional constant. Finally, since clearly

4.3) in [41]. Since
n R −α for some dimensional constant c ′′ n ≥ c ′ n , the conclusion follows. With Lemma 4.1 at our disposal, we can prove the following result.

Remark 4.3 (H 1 − W α,1 interpolation inequality). Thanks to
for all R > 0. Hence for all R > 0, and the desired inequality follows by optimizing the parameter R > 0 in the right-hand side.
(i) For all given p ∈ (1, +∞), the operator in (4.11) uniquely extends to a bounded linear operator T m α,β : The operator in (4.11) uniquely extends to a bounded linear operator T m α,β :
Proof. Without loss of generality, we can directly assume that 0 ≤ γ < β < α ≤ 1. We prove the two statements separately.
Proof of (i). Given f ∈ S α,p (R n ), we can write thanks to Lemma 4.4(i). By performing a dilation and by optimizing the right-hand side, we find that because f ∈ L α+γ,p (R n ) and by the L p -continuity property of the Riesz transform, we get that ∇ γ f ∈ S α,p (R n ; R n ) according to the definition given in (2.7) and the identification established in Corollary 2.1. Repeating the above computations for (each component of) the function ∇ γ f ∈ S α,p (R n ; R n ) with exponents α − γ and β − γ in place of α and β respectively and then optimizing, we get where c n,p = σ n n 1/2p max p, 1 p−1 . Thanks to Theorem A.1, Proposition B.1 and Proposition B.4, inequality (4.12) follows by performing a standard approximation argument.
In the case γ = 0, inequality (4.13) follows from (4.12) by the L p -continuity of the Riesz transform. This concludes the proof of (i).
Proof of (ii). Given f ∈ HS α,1 (R n ), arguing as above, we can write thanks to Lemma 4.4(ii). By performing a dilation and by optimising the right-hand side, we find that by Proposition 3.4(ii). Moreover, because f ∈ HS α+γ,1 (R n ) and by the H 1 -continuity property of the Riesz transform.
Thus ∇ γ f ∈ HS α,1 (R n ; R n ). Repeating the above computations for (each component of) the function ∇ γ f ∈ HS α,1 (R n ; R n ) with exponents α − γ and β − γ in place of α and β respectively and then optimizing, we get where c n = σ n n 1/2 . Thanks to Lemma 2.3, inequality (4.14) follows by performing a standard approximation argument.
In the case γ = 0, inequality (4.15) follows from (4.12) by the H 1 -continuity of the Riesz transform. This concludes the proof of (ii).

Asymptotic behavior of fractional α-variation as α → 0 +
In this section, we study the asymptotic behavior of ∇ α as α → 0 + .
For β ∈ (0, α), the operator and where c n,p is as in (5.2). Finally, if p < +∞ and f ∈ C 0,α loc (R n ) ∩ L p (R n ), then ∇ 0 f is well defined and belongs to C 0 (R n ; R n ), (5.3) holds for β = 0, for all bounded open sets U ⊂ R n we have
Proof. We divide the proof in four steps.

Remark 5.2.
It is easy to see that a result analogous to Lemma 5.1 can be proved for the fractional divergence operator. In particular, if ϕ ∈ C 0,α (R n ; R n ) ∩ L p (R n ; R n ) for some α ∈ (0, 1] and p ∈ [1, +∞], then div β ϕ ∈ L ∞ (R n ) for all β ∈ (0, α) with where c n,p > 0 is the constant defined in (5.2). If p < +∞, then div β ϕ ∈ L ∞ (R n ) for all β ∈ [0, α), the above estimate holds also for β = 0 and we have As an immediate consequence of Lemma 5.1 and Remark 5.2, we can show that the fractional α-variation is lower semicontinuous as α → 0 + .

5.2.
Strong and energy convergence of ∇ α as α → 0 + . We now study the strong and the energy convergence of ∇ α as α → 0 + . For the strong convergence, we have the following result.
(i) If f ∈ α∈(0,1) HS α,1 (R n ), then Remark 5.5. Thanks to Corollary 3.5, Theorem 5.4(i) can be equivalently stated as We prove Theorem 5.4 in Section 5.3. For the convergence of the (rescaled) energy, we instead have the following result.

Proof of Theorem 5.4.
Before the proof of Theorem 5.4, we need to recall the following well-known result, see the first part of the proof of [37,Lemma 1.60]. For the reader's convenience and to keep the paper as self-contained as possible, we briefly recall its simple proof.
Proof. By means of the Fourier transform, the problem can be equivalently restated as follows: if ϕ ∈ S(R n ) satisfies ∂ a ϕ(0) = 0 for all a ∈ N n 0 such that |a| ≤ m, then ϕ(ξ) = n 1 ξ i ψ i (ξ) for some ψ 1 , . . . , ψ n ∈ S(R n ) with ∂ a ψ i (0) = 0 for all i = 1, . . . , n and all a ∈ N n 0 such that |a| ≤ m − 1. This can be achieved as follows. Fixed any ζ ∈ C ∞ c (R n ) such that supp ζ ⊂ B 2 and ζ ≡ 1 on B 1 , we can define for all i = 1, . . . , n. It is now easy to prove that such ψ i 's satisfy the required properties and we leave the simple calculations to the reader.
Thanks to Lemma 5.7, we can prove the following L p -convergence result of the fractional α-Laplacian of suitably regular functions as α → 0 + , as well as analogous convergence results for the fractional α-gradient.
As a consequence, if p ∈ (1, +∞) and f ∈ S 0 (R n ), then Proof. Let f ∈ S 0 (R n ) be fixed. If p ∈ (1, +∞), then by the L p -continuity of the Riesz transform, so that (5.11) is a consequence of (5.10). To prove (5.10), given x ∈ R n we write is the constant appearing in (2.4). One easily sees that On the one hand, we can estimate Integrating by parts, the reader can easily verify that for all p ∈ [1, +∞]. Hence we get for all p ∈ [1, +∞], so that we obtain (5.10) and (5.11). Finally, let f ∈ S ∞ (R n ), so that Rf ∈ S 0 (R n ; R n ), R(Rf ) ∈ S 0 (R n ; R n 2 ) and (−∆) and thus lim α→0 + ∇ α f − Rf H 1 (R n ; R n ) = 0 thanks (5.10) (which clearly holds also for vector-valued functions). Thus, we obtain (5.12), and the proof is complete.
We can now prove Theorem 5.4.
Proof of Theorem 5.4. We prove the two statements separately.
Proof of (i). Let f ∈ HS α,1 (R n ). By Lemma 2.3, there exists (f k ) k∈N ⊂ S ∞ (R n ) such that f k → f in HS α,1 (R n ) as k → +∞. If β ∈ (0, α), then we can estimate (4.15) in Theorem 4.5(ii) and the H 1 -continuity of the Riesz transform, where c n , c ′ n > 0 are dimensional constants. Thus lim sup (5.12) in Lemma 5.8, where c ′′ n = c n + c ′ n . Hence (5.7) follows by passing to the limit as k → +∞ and the proof of (i) is complete.
Proof of (ii). We argue as in the proof of (i). Let f ∈ S α,p (R n ). By Proposition 2.2, (4.13) in Theorem 4.5(i) and the L p -continuity of the Riesz transform, where the constants c n,p , c ′ n,p > 0 depend only on n and p. Thus lim sup (5.11) in Lemma 5.8, where c ′′ n,p = c n,p + c ′ n,p . Hence (5.8) follows by passing to the limit as k → +∞ and the proof of (ii) is complete. Remark 5.9 (Direct proof of (1.14)). The proof of (1.14), i.e., immediately follows from Theorem 5.4(i) and Remark 5.5. As briefly discussed in Section 1.3, one can directly prove (1.14) by combining the interpolation inequality proven in Theorem 4.2 with an approximation argument as done in the proof of Theorem 5.4. We let the interested reader fill the easy details. 5.4. Proof of Theorem 5.6. We now pass to the proof of Theorem 5.6. We need some preliminaries. We begin with the following result. Lemma 5.10. Let f ∈ L 1 (R n ) and let R ∈ (0, +∞) be such that supp f ⊂ B R . If ε > R, then Proof. Since µ n,α → µ n,0 as α → 0 + , we just need to prove that We now divide the proof in two steps.
Thanks to Lemma 5.10, we can prove the following result.
We are now ready to prove Theorem 5.6.
In the following result, we recall the self-adjointness property the fractional Laplacian.
The following result provides an L p -estimate on translations of functions in S α,p (R n ). It can be stated by saying that the inclusion S α,p (R n ) ⊂ B α p,∞ (R n ) is continuous, where B α p,q (R n ) is the Besov space, see [46,Chapter 14]. For a similar result in the W α,p (R n ) space, we refer the reader to [30].
Thanks to Corollary 2.1, this result can be derived from the analogous result already known for functions in L α,p (R n ). However, the estimate in (B.2) provides an explicit constant (independent of p) that may be of some interest. The proof of Proposition B.2 below can be easily established following the one of [27, Proposition 3.14](and exploiting Minkowski's integral inequality and Theorem A.1) and we leave it to the reader. Proposition B.2. Let α ∈ (0, 1) and p ∈ [1, +∞). If f ∈ S α,p (R n ), then for all y ∈ R n , where γ n,α > 0 is as in [27,Proposition 3.14].
A similar result holds for spaces BV α (R n ), indeed from [27, Proposition 3.14], one immediately deduces that the inclusion BV α (R n ) ⊂ B α 1,∞ (R n ) holds continuously for all α ∈ (0, 1). The next result shows that this inclusion is actually strict whenever n ≥ 2.
Proof. By [27, Theorem 3.9], we just need to prove that B α 1,∞ (R n ) \ L n n−α (R n ) = ∅. Let η 1 ∈ C ∞ c (R n ) be as in (4.1) and (4.2), and let f (x) = η 1 (x)|x| α−n for all x ∈ R n . On the one side, we clearly have f / ∈ L n n−α (R n ). On the other side, for all h ∈ R n with |h| < 1, we can estimate where C > 0 is a constant depending only on n and α (that may vary from line to line). Thus f ∈ B α 1,∞ (R n ) and the conclusion follows. We conclude with the following result which, again, can be derived from the theory of Bessel potential spaces. We state it here since our distributional approach provides explicit constants (independent of p) in the estimates that may be of some interest. The proof is very similar to the one of [28, Proposition 3.12] and we leave it to the interested reader.