A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
aa r X i v : . [ m a t h . F A ] O c t A DISTRIBUTIONAL APPROACH TO FRACTIONAL SOBOLEVSPACES AND FRACTIONAL VARIATION: ASYMPTOTICS I
GIOVANNI E. COMI AND GIORGIO STEFANI
Abstract.
We continue the study of the space BV α ( R n ) of functions with boundedfractional variation in R n of order α ∈ (0 ,
1) introduced in our previous work [10], bydealing with the asymptotic behaviour of the fractional operators involved. After sometechnical improvements of certain results of [10], we prove that the fractional α -variationconverges to the standard De Giorgi’s variation both pointwise and in the Γ-limit senseas α → − . We also prove that the fractional β -variation converges to the fractional α -variation both pointwise and in the Γ-limit sense as β → α − for any given α ∈ (0 , Contents
1. Introduction 21.1. A distributional approach to fractional variation 21.2. Asymptotics and Γ-convergence in the standard fractional setting 41.3. Asymptotics and Γ-convergence for the fractional α -variation as α → − α -variation as α → + ∇ α and div α ∇ α and div α to Lip b -regular tests 112.4. Extended Leibniz’s rules for ∇ α and div α W α, and BV α seminorms 163. Estimates and representation formulas for the fractional α -gradient 203.1. Integrability properties of the fractional α -gradient 203.2. Two representation formulas for the α -variation 273.3. Relation between BV β and BV α,p for β < α and p > BV α ⊂ W β, for β < α : a representation formula 324. Asymptotic behaviour of fractional α -variation as α → − ∇ α and div α as α → − α -variation as α → − Date : October 30, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Function with bounded fractional variation, fractional perimeter, fractionalcalculus, fractional derivative, fractional gradient, fractional divergence, Gamma-convergence.
Acknowledgements . The authors thank Luigi Ambrosio, Elia Brué, Mattia Calzi, Quoc-Hung Nguyenand Daniel Spector for many valuable suggestions and useful comments. This research was partiallysupported by the PRIN2015 MIUR Project “Calcolo delle Variazioni”. α -variation as α → − β -variation as β → α − ∇ β and div β as β → α β -variation as β → α − β -variation as β → α − BV functions 53A.1. Truncation of BV functions 53A.2. Approximation by sets with polyhedral boundary 55References 591. Introduction
A distributional approach to fractional variation.
In our previous work [10],we introduced the space BV α ( R n ) of functions with bounded fractional variation in R n of order α ∈ (0 , f ∈ L ( R n ) belongs to the space BV α ( R n ) if its fractional α -variation | D α f | ( R n ) := sup (cid:26)Z R n f div α ϕ dx : ϕ ∈ C ∞ c ( R n ; R n ) , k ϕ k L ∞ ( R n ; R n ) ≤ (cid:27) (1.1)is finite. Here div α ϕ ( x ) := µ n,α Z R n ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 dy, x ∈ R n , (1.2)is the fractional α -divergence of ϕ ∈ C ∞ c ( R n ; R n ), where µ n,α := 2 α π − n Γ (cid:16) n + α +12 (cid:17) Γ (cid:16) − α (cid:17) (1.3)for any given α ∈ (0 , α was introduced in [35] as the natural dual operator of the much more studied fractional α -gradient ∇ α f ( x ) := µ n,α Z R n ( y − x )( f ( y ) − f ( x )) | y − x | n + α +1 dy, x ∈ R n , (1.4)defined for all f ∈ C ∞ c ( R n ). For an account on the existing literature on the operator ∇ α ,see [31, Section 1]. Here we only refer to [29–33, 35–37] for the articles tightly connectedto the present work and to [27, Section 15.2] for an agile presentation of the fractional op-erators defined in (1.2) and in (1.4) and of some of their elementary properties. Accordingto [33, Section 1], it is interesting to notice that [20] seems to be the earliest reference forthe operator defined in (1.4).The operators in (1.2) and in (1.4) are dual in the sense that Z R n f div α ϕ dx = − Z R n ϕ · ∇ α f dx (1.5)for all f ∈ C ∞ c ( R n ) and ϕ ∈ C ∞ c ( R n ; R n ), see [35, Section 6] and [10, Lemma 2.5].Moreover, both operators have good integrability properties when applied to test func-tions, namely ∇ α f ∈ L p ( R n ) and div α ϕ ∈ L p ( R n ; R n ) for all p ∈ [1 , + ∞ ] for any given f ∈ C ∞ c ( R n ) and ϕ ∈ C ∞ c ( R n ; R n ), see [10, Corollary 2.3]. DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 3
The integration-by-part formula (1.5) represents the starting point for the distributionalapproach to fractional Sobolev spaces and fractional variation we developed in [10]. Infact, similarly to the classical case, a function f ∈ L ( R n ) belongs to BV α ( R n ) if andonly if there exists a finite vector-valued Radon measure D α f ∈ M ( R n ; R n ) such that Z R n f div α ϕ dx = − Z R n ϕ · dD α f (1.6)for all ϕ ∈ C ∞ c ( R n ; R n ), see [10, Theorem 3.2].Motivated by (1.6) and similarly to the classical case, we can define the weak fractional α -gradient of a function f ∈ L p ( R n ), with p ∈ [1 , + ∞ ], as the function ∇ αw f ∈ L ( R n ; R n )satisfying Z R n f div α ϕ dx = − Z R n ∇ αw f · ϕ dx for all ϕ ∈ C ∞ c ( R n ; R n ). For α ∈ (0 ,
1) and p ∈ [1 , + ∞ ], we can thus define the distribu-tional fractional Sobolev space S α,p ( R n ) := { f ∈ L p ( R n ) : ∃ ∇ αw f ∈ L p ( R n ; R n ) } (1.7)naturally endowed with the norm k f k S α,p ( R n ) := k f k L p ( R n ) + k∇ αw f k L p ( R n ; R n ) ∀ f ∈ S α,p ( R n ) . (1.8)It is interesting to compare the distributional fractional Sobolev spaces S α,p ( R n ) withthe well-known fractional Sobolev space W α,p ( R n ), that is, the space W α,p ( R n ) := f ∈ L p ( R n ) : [ f ] W α,p ( R n ) := Z R n Z R n | f ( x ) − f ( y ) | p | x − y | n + pα dx dy ! p < + ∞ endowed with the norm k f k W α,p ( R n ) := k f k L p ( R n ) + [ f ] W α,p ( R n ) ∀ f ∈ W α,p ( R n ) . If p = + ∞ , then W α, ∞ ( R n ) naturally coincides with the space of bounded α -Höldercontinuous functions endowed with the usual norm (see [14] for a detailed account on thespaces W α,p ).For the case p = 1, starting from the very definition of the fractional gradient ∇ α ,it is plain to see that W α, ( R n ) ⊂ S α, ( R n ) ⊂ BV α ( R n ) with both (strict) continuousembeddings, see [10, Theorems 3.18 and 3.25].For the case p ∈ (1 , + ∞ ), instead, it is known that S α,p ( R n ) ⊃ L α,p ( R n ) with continuousembedding, where L α,p ( R n ) is the Bessel potential space of parameters α ∈ (0 ,
1) and p ∈ (1 , + ∞ ), see [10, Section 3.9] and the references therein. In the forthcoming paper [9],it will be proved that also the inclusion S α,p ( R n ) ⊂ L α,p ( R n ) holds continuously, so thatthe spaces S α,p ( R n ) and L α,p ( R n ) coincide. In particular, we get the following relations: S α + ε,p ( R n ) ⊂ W α,p ( R n ) ⊂ S α − ε,p ( R n ) with continuous embeddings for all α ∈ (0 , p ∈ (1 , + ∞ ) and 0 < ε < min { α, − α } , see [32, Theorem 2.2]; S α, ( R n ) = W α, ( R n ) forall α ∈ (0 , W α,p ( R n ) ⊂ S α,p ( R n ) with continuous embedding forall α ∈ (0 ,
1) and p ∈ (1 , geometric regime p = 1, our distributional approach to the fractional variationnaturally provides a new definition of distributional fractional perimeter. Precisely, for G. E. COMI AND G. STEFANI any open set Ω ⊂ R n , the fractional Caccioppoli α -perimeter in Ω of a measurable set E ⊂ R n is the fractional α -variation of χ E in Ω, i.e. | D α χ E | (Ω) = sup (cid:26)Z E div α ϕ dx : ϕ ∈ C ∞ c (Ω; R n ) , k ϕ k L ∞ (Ω; R n ) ≤ (cid:27) . Thus, E is a set with finite fractional Caccioppoli α -perimeter in Ω if | D α χ E | (Ω) < + ∞ .Similarly to the aforementioned embedding W α, ( R n ) ⊂ BV α ( R n ), we have the in-equality | D α χ E | (Ω) ≤ µ n,α P α ( E ; Ω) (1.9)for any open set Ω ⊂ R n , see [10, Proposition 4.8], where P α ( E ; Ω) := Z Ω Z Ω | χ E ( x ) − χ E ( y ) || x − y | n + α dx dy + 2 Z Ω Z R n \ Ω | χ E ( x ) − χ E ( y ) || x − y | n + α dx dy (1.10)is the standard fractional α -perimeter of a measurable set E ⊂ R n relative to the open setΩ ⊂ R n (see [11] for an account on the fractional perimeter P α ). Note that, by definition,the fractional α -perimeter of E in R n is simply P α ( E ) := P α ( E ; R n ) = [ χ E ] W α, ( R n ) . Weremark that inequality (1.9) is strict in most of the cases, as shown in Section 2.6 below.This completely answers a question left open in our previous work [10].1.2. Asymptotics and Γ -convergence in the standard fractional setting. Thefractional Sobolev space W α,p ( R n ) can be understood as an ‘intermediate space’ betweenthe space L p ( R n ) and the standard Sobolev space W ,p ( R n ). In fact, W α,p ( R n ) can berecovered as a suitable (real) interpolation space between the spaces L p ( R n ) and W ,p ( R n ).We refer to [5, 40] for a general introduction on interpolation spaces and to [26] for a morespecific treatment of the interpolation space between L p ( R n ) and W ,p ( R n ).One then naturally expects that, for a sufficiently regular function f , the fractionalSobolev seminorm [ f ] W α,p ( R n ) , multiplied by a suitable renormalising constant, shouldtend to k f k L p ( R n ) as α → + and to k∇ f k L p ( R n ) as α → − . Indeed, for p ∈ [1 , + ∞ ) and α ∈ (0 , α → + α [ f ] pW α,p ( R n ) = A n,p k f k pL p ( R n ) (1.11)for all f ∈ S α ∈ (0 , W α,p ( R n ), whilelim α → − (1 − α ) [ f ] pW α,p ( R n ) = B n,p k∇ f k pL p ( R n ) (1.12)for all f ∈ W ,p ( R n ). Here A n,p , B n,p > n, p . Thelimit (1.11) was proved in [23, 24], while the limit (1.12) was established in [6]. As provedin [13], when p = 1 the limit (1.12) holds in the more general case of BV functions, thatis, lim α → − (1 − α ) [ f ] W α, ( R n ) = B n, | Df | ( R n ) (1.13)for all f ∈ BV ( R n ). For a different approach to the limits in (1.11) and in (1.13) basedon interpolation techniques, see [26].Concerning the fractional perimeter P α given in (1.10), one has some additional infor-mation besides equations (1.11) and (1.13).On the one hand, thanks to [28, Theorem 1.2], the fractional α -perimeter P α enjoys thefollowing fractional analogue of Gustin’s Boxing Inequality (see [19] and [16, Corollary
DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 5 c n > E ⊂ R n , one can find a covering E ⊂ [ k ∈ N B r k ( x k )of open balls such that X k ∈ N r n − αk ≤ c n α (1 − α ) P α ( E ) . (1.14)Inequality (1.14) bridges the two limiting behaviours given by (1.11) and (1.13) andprovides a useful tool for recovering Gagliardo–Nirenberg–Sobolev and Poincaré–Sobolevinequalities that remain stable as the exponent α ∈ (0 ,
1) approaches the endpoints.On the other hand, by [2, Theorem 2], the fractional α -perimeter P α Γ-converges in L ( R n ) to the standard De Giorgi’s perimeter P as α → − , that is, if Ω ⊂ R n is abounded open set with Lipschitz boundary, thenΓ( L ) - lim α → − (1 − α ) P α ( E ; Ω) = 2 ω n − P ( E ; Ω) (1.15)for all measurable sets E ⊂ R n , where ω n is the volume of the unit ball in R n (it shouldbe noted that in [2] the authors use a slightly different definition of the fractional α -perimeter, since they consider the functional J α ( E, Ω) := P α ( E, Ω)). For a completeaccount on Γ -convergence , we refer the reader to the monographs [7, 12] (throughout allthe paper, with the symbol Γ( X ) - lim we denote the Γ-convergence in the ambient metricspace X ). The convergence in (1.15), besides giving a Γ-convergence analogue of the limitin (1.13), is tightly connected with the study of the regularity properties of non-localminimal surfaces , that is, (local) minimisers of the fractional α -perimeter P α .1.3. Asymptotics and Γ -convergence for the fractional α -variation as α → − . The main aim of the present work is to study the asymptotic behaviour of the fractional α -variation (1.1) as α → − , both in the pointwise and in the Γ-convergence sense.We provide counterparts of the limits (1.12) and (1.13) for the fractional α -variation.Indeed, we prove that, if f ∈ W ,p ( R n ) for some p ∈ (1 , + ∞ ), then f ∈ S α,p ( R n ) for all α ∈ (0 ,
1) and, moreover, lim α → − k∇ αw f − ∇ w f k L p ( R n ; R n ) = 0 . (1.16)In the geometric regime p = 1, we show that if f ∈ BV ( R n ) then f ∈ BV α ( R n ) for all α ∈ (0 ,
1) and, in addition, D α f ⇀ Df in M ( R n ; R n ) and | D α f | ⇀ | Df | in M ( R n ) as α → − (1.17)and lim α → − | D α f | ( R n ) = | Df | ( R n ) . (1.18)We are also able to treat the case p = + ∞ . In fact, we prove that if f ∈ W , ∞ ( R n ) then f ∈ S α, ∞ ( R n ) for all α ∈ (0 ,
1) and, moreover, ∇ αw f ⇀ ∇ w f in L ∞ ( R n ; R n ) as α → − (1.19)and k∇ w f k L ∞ ( R n ; R n ) ≤ lim inf α → − k∇ αw f k L ∞ ( R n ; R n ) . (1.20) G. E. COMI AND G. STEFANI
We refer the reader to Theorem 4.9, Theorem 4.10 and Theorem 4.11 below for the precisestatements. We warn the reader that the symbol ‘ ⇀ ’ appearing in (1.17) and (1.19)denotes the weak*-convergence , see Section 2.1 below for the notation.Some of the above results were partially announced in [34]. In a similar perspective,we also refer to the work [25], where the authors proved convergence results for non-localgradient operators on BV functions defined on bounded open sets with smooth boundary.The approach developed in [25] is however completely different from the asymptotic analy-sis we presently perform for the fractional operator defined in (1.4), since the boundednessof the domain of definition of the integral operators considered in [25] plays a crucial role.Notice that the renormalising factor (1 − α ) p is not needed in the limits (1.16) – (1.20),contrarily to what happened for the limits (1.12) and (1.13). In fact, this difference shouldnot come as a surprise, since the constant µ n,α in (1.3), encoded in the definition of theoperator ∇ α , satisfies µ n,α ∼ − αω n as α → − , (1.21)and thus plays a similar role of the factor (1 − α ) p in the limit as α → − . Thus,differently from our previous work [10], the constant µ n,α appearing in the definition ofthe operators ∇ α and div α is of crucial importance in the asymptotic analysis developedin the present paper.Another relevant aspect of our approach is that convergence as α → − holds true notonly for the total energies, but also at the level of differential operators, in the strongsense when p ∈ (1 , + ∞ ) and in the weak* sense for p = 1 and p = + ∞ . In simpler terms,the non-local fractional α -gradient ∇ α converges to the local gradient ∇ as α → − in themost natural way every time the limit is well defined.We also provide a counterpart of (1.15) for the fractional α -variation as α → − .Precisely, we prove that, if Ω ⊂ R n is a bounded open set with Lipschitz boundary, thenΓ( L ) - lim α → − | D α χ E | (Ω) = P ( E ; Ω) (1.22)for all measurable set E ⊂ R n , see Theorem 4.16. In view of (1.9), one may askwhether the Γ - lim sup inequality in (1.22) could be deduced from the Γ - lim sup in-equality in (1.15). In fact, by employing (1.9) together with (1.15) and (1.21), one canestimateΓ( L ) - lim sup α → − | D α χ E | (Ω) ≤ Γ( L ) - lim sup α → − µ n,α P α ( E, Ω) = 2 ω n − ω n P ( E, Ω) . However, we have ω n − ω n > n ≥ n = 1. In a similar way,one sees that the Γ - lim inf inequality in (1.22) implies the Γ - lim inf inequality in (1.15)only in the case n = 1.Besides the counterpart of (1.15), our approach allows to prove that Γ-convergenceholds true also at the level of functions. Indeed, if f ∈ BV ( R n ) and Ω ⊂ R n is an openset such that either Ω is bounded with Lipschitz boundary or Ω = R n , thenΓ( L ) - lim α → − | D α f | (Ω) = | Df | (Ω) . (1.23) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 7
We refer the reader to Theorem 4.13, Theorem 4.14 and Theorem 4.17 for the (even moregeneral) results in this direction. Again, similarly as before and thanks to the asymp-totic behaviour (1.21), the renormalising factor (1 − α ) is not needed in the limits (1.22)and (1.23).As a byproduct of the techniques developed for the asymptotic study of the fractional α -variation as α → − , we are also able to characterise the behaviour of the fractional β -variation as β → α − , for any given α ∈ (0 , f ∈ BV α ( R n ), then D β f ⇀ D α f in M ( R n ; R n ) and | D β f | ⇀ | D α f | in M ( R n ) as β → α − and, moreover, lim β → α − | D β f | ( R n ) = | D α f | ( R n ) , see Theorem 5.4. On the other hand, if f ∈ BV α ( R n ) and Ω ⊂ R n is an open set suchthat either Ω is bounded and | D α f | ( ∂ Ω) = 0 or Ω = R n , thenΓ( L ) - lim β → α − | D β f | (Ω) = | D α f | (Ω) , see Theorem 5.5 and Theorem 5.6.1.4. Future developments: asymptotics for the fractional α -variation as α → + . Having in mind the limit (1.11), it would be interesting to understand what happens tothe fractional α -variation (1.1) as α → + . Note thatlim α → + µ n,α = π − n Γ (cid:16) n +12 (cid:17) Γ (cid:16) (cid:17) =: µ n, , (1.24)so there is no renormalisation factor as α → + , differently from (1.21).At least formally, as α → + the fractional α -gradient in (1.4) is converging to theoperator ∇ f ( x ) := µ n, Z R n ( y − x )( f ( y ) − f ( x )) | y − x | n +1 dy, x ∈ R n . (1.25)The operator in (1.25) is well defined for all f ∈ C ∞ c ( R n ) and, actually, coincides with thewell-known vector-valued Riesz transform Rf , see [17, Section 5.1.4] and [38, Chapter 3].Similarly, the fractional α -divergence in (1.2) is formally converging to the operatordiv ϕ ( x ) := µ n, Z R n ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n +1 dy, x ∈ R n , (1.26)which is well defined for all ϕ ∈ C ∞ c ( R n ; R n ).In perfect analogy with what we did before, we can introduce the space BV ( R n ) asthe space of functions f ∈ L ( R n ) such that the quantity | D f | ( R n ) := sup (cid:26)Z R n f div ϕ dx : ϕ ∈ C ∞ c ( R n ; R n ) , k ϕ k L ∞ ( R n ; R n ) ≤ (cid:27) is finite. Surprisingly (and differently from the fractional α -variation, recall [10, Sec-tion 3.10]), it turns out that | D f | ≪ L n for all f ∈ BV ( R n ). More precisely, one canactually prove that BV ( R n ) = H ( R n ), in the sense that f ∈ BV ( R n ) if and only if f ∈ H ( R n ), with D f = Rf L n in M ( R n ; R n ). Here H ( R n ) := n f ∈ L ( R n ) : Rf ∈ L ( R n ; R n ) o G. E. COMI AND G. STEFANI is the (real)
Hardy space , see [39, Chapter III] for the precise definition. Thus, it wouldbe interesting to understand for which functions f ∈ L ( R n ) the fractional α -gradient ∇ α f tends (in a suitable sense) to the Riesz transform Rf as α → + . Of course, if Rf / ∈ L ( R n ; R n ), that is, f / ∈ H ( R n ), then one cannot expect strong convergence in L and, instead, may consider the asymptotic behaviour of the rescaled fractional gradient α ∇ α f as α → + , in analogy with the limit in (1.11). This line of research, as well asthe identifications BV = H and S α,p = L α,p mentioned above, will be the subject of theforthcoming paper [9].1.5. Organisation of the paper.
The paper is organised as follows. In Section 2,after having briefly recalled the definitions and the main properties of the operators ∇ α and div α , we extend certain technical results of [10]. In Section 3, we prove severalintegrability properties of the fractional α -gradient and two useful representation formulasfor the fractional α -variation of functions with bounded De Giorgi’s variation. We arealso able to prove similar results for the fractional β -gradient of functions with boundedfractional α -variation, see Section 3.4. In Section 4, we study the asymptotic behaviour ofthe fractional α -variation as α → − and prove pointwise-convergence and Γ-convergenceresults, dealing separately with the integrability exponents p = 1, p ∈ (1 , + ∞ ) and p = + ∞ . In Section 5, we show that the fractional β -variation weakly converges and Γ-converges to the fractional α -variation as β → α − for any α ∈ (0 , BV functions and sets with finite perimeter that are used in Section 3and in Section 4. 2. Preliminaries
General notation.
We start with a brief description of the main notation used inthis paper. In order to keep the exposition the most reader-friendly as possible, we retainthe same notation adopted in our previous work [10].Given an open set Ω, we say that a set E is compactly contained in Ω, and we write E ⋐ Ω, if the E is compact and contained in Ω. We denote by L n and H α the n -dimensional Lebesgue measure and the α -dimensional Hausdorff measure on R n respec-tively, with α ≥
0. Unless otherwise stated, 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 areLebesgue measurable, unless otherwise stated. We denote by B r ( x ) the standard openEuclidean ball with center x ∈ R n and radius r >
0. We let B r = B r (0). Recall that ω n := | B | = π n / Γ (cid:16) n +22 (cid:17) and H n − ( ∂B ) = nω n , where Γ is Euler’s Gamma function ,see [4].We let GL( n ) ⊃ O( n ) ⊃ SO( n ) be the general linear group , the orthogonal group andthe special orthogonal group respectively. We tacitly identify GL( n ) ⊂ R n with the spaceof invertible n × n - matrices and we endow it with the usual Euclidean distance in R n .For k ∈ N ∪ { + ∞} and m ∈ N , we denote by C kc (Ω; R m ) and Lip c (Ω; R m ) the spaces of C k -regular and, respectively, Lipschitz-regular, m -vector-valued functions defined on R n with compact support in Ω.For any exponent p ∈ [1 , + ∞ ], we denote by L p (Ω; R m ) := n u : Ω → R m : k u k L p (Ω; R m ) < + ∞ o DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 9 the space of m -vector-valued Lebesgue p -integrable functions on Ω. For p ∈ [1 , + ∞ ], wesay that ( f k ) k ∈ N ⊂ L p (Ω; R m ) weakly converges to f ∈ L p (Ω; R m ), and we write f k ⇀ f in L p (Ω; R m ) as k → + ∞ , if lim k → + ∞ Z Ω f k · ϕ dx = Z Ω f · ϕ dx (2.1)for all ϕ ∈ L q (Ω; R m ), with q ∈ [1 , + ∞ ] the conjugate exponent of p , that is, p + q = 1 (withthe usual convention ∞ = 0). Note that in the case p = + ∞ we make a little abuse ofterminology, since the limit in (2.1) actually defines the weak*-convergence in L ∞ (Ω; R m ).We denote by W ,p (Ω; R m ) := n u ∈ L p (Ω; R m ) : [ u ] W ,p (Ω; R m ) := k∇ u k L p (Ω; R n + m ) < + ∞ o the space of m -vector-valued Sobolev functions on Ω, see for instance [21, Chapter 10] forits precise definition and main properties. We also let w ,p (Ω; R m ) := n u ∈ L (Ω; R m ) : [ u ] W ,p (Ω; R m ) < + ∞ o . We denote by BV (Ω; R m ) := n u ∈ L (Ω; R m ) : [ u ] BV (Ω; R m ) := | Du | (Ω) < + ∞ o the space of m -vector-valued functions of bounded variation on Ω, see for instance [3,Chapter 3] or [15, Chapter 5] for its precise definition and main properties. We also let bv (Ω; R m ) := n u ∈ L (Ω; R m ) : [ u ] BV (Ω; R m ) < + ∞ o . For α ∈ (0 ,
1) and p ∈ [1 , + ∞ ), we denote by W α,p (Ω; R m ) := u ∈ L p (Ω; R m ) : [ u ] W α,p (Ω; R m ) := Z Ω Z Ω | u ( x ) − u ( y ) | p | x − y | n + pα dx dy ! p < + ∞ the space of m -vector-valued fractional Sobolev functions on Ω, see [14] for its precisedefinition and main properties. We also let w α,p (Ω; R m ) := n u ∈ L (Ω; R m ) : [ u ] W α,p (Ω; R m ) < + ∞ o . For α ∈ (0 ,
1) and p = + ∞ , we simply let W α, ∞ (Ω; R m ) := ( u ∈ L ∞ (Ω; R m ) : sup x,y ∈ Ω , x = y | u ( x ) − u ( y ) || x − y | α < + ∞ ) , so that W α, ∞ (Ω; R m ) = C ,αb (Ω; R m ), the space of m -vector-valued bounded α -Höldercontinuous functions on Ω.We let M (Ω; R m ) be the space of m -vector-valued Radon measures with finite totalvariation, precisely | µ | (Ω) := sup (cid:26)Z Ω ϕ · dµ : ϕ ∈ C c (Ω; R m ) , k ϕ k L ∞ (Ω; R m ) ≤ (cid:27) for µ ∈ M (Ω; R m ). We say that ( µ k ) k ∈ N ⊂ M (Ω; R m ) weakly converges to µ ∈ M (Ω; R m ),and we write µ k ⇀ µ in M (Ω; R m ) as k → + ∞ , iflim k → + ∞ Z Ω ϕ · dµ k = Z Ω ϕ · dµ (2.2) for all ϕ ∈ C c (Ω; R m ). Note that we make a little abuse of terminology, since the limitin (2.2) actually defines the weak*-convergence in M (Ω; R m ).In order to avoid heavy notation, if the elements of a function space F (Ω; R m ) arereal-valued (i.e. m = 1), then we will drop the target space and simply write F (Ω).2.2. Basic properties of ∇ α and div α . We recall the non-local operators ∇ α and div α introduced by Šilhavý in [35] (see also our previous work [10]).Let α ∈ (0 ,
1) and set µ n,α := 2 α π − n Γ (cid:16) n + α +12 (cid:17) Γ (cid:16) − α (cid:17) . We let ∇ α f ( x ) := µ n,α lim ε → Z {| z | >ε } zf ( x + z ) | z | n + α +1 dz be the fractional α -gradient of f ∈ Lip c ( R n ) at x ∈ R n . We also letdiv α ϕ ( x ) := µ n,α lim ε → Z {| z | >ε } z · ϕ ( x + z ) | z | n + α +1 dz be the fractional α -divergence of ϕ ∈ Lip c ( R n ; R n ) at x ∈ R n . The non-local operators ∇ α and div α are well defined in the sense that the involved integrals converge and the limitsexist, see [35, Section 7] and [10, Section 2]. Moreover, since Z {| z | >ε } z | z | n + α +1 dz = 0 , ∀ ε > , it is immediate to check that ∇ α c = 0 for all c ∈ R and ∇ α f ( x ) = µ n,α lim ε → Z {| y − x | >ε } ( y − x ) | y − x | n + α +1 f ( y ) dy = µ n,α lim ε → Z {| x − y | >ε } ( y − x )( f ( y ) − f ( x )) | y − x | n + α +1 dy = µ n,α Z R n ( y − x )( f ( y ) − f ( x )) | y − x | n + α +1 dy, ∀ x ∈ R n , for all f ∈ Lip c ( R n ). Analogously, we also havediv α ϕ ( x ) = µ n,α lim ε → Z {| x − y | >ε } ( y − x ) · ϕ ( y ) | y − x | n + α +1 dy, = µ n,α lim ε → Z {| x − y | >ε } ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 dy, = µ n,α Z R n ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 dy, ∀ x ∈ R n , for all ϕ ∈ Lip c ( R n ).Given α ∈ (0 , n ), we let I α f ( x ) := µ n, − α n − α Z R n u ( y ) | x − y | n − α dy, x ∈ R n , (2.3) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 11 be the
Riesz potential of order α ∈ (0 , n ) of a function u ∈ C ∞ c ( R n ; R m ). We recall that,if α, β ∈ (0 , n ) satisfy α + β < n , then we have the following semigroup property I α ( I β u ) = I α + β u (2.4)for all u ∈ C ∞ c ( R n ; R m ). In addition, if 1 < p < q < + ∞ satisfy1 q = 1 p − αn , then there exists a constant C n,α,p > k I α u k L q ( R n ; R m ) ≤ C n,α,p k u k L p ( R n ; R m ) (2.5)for all u ∈ C ∞ c ( R n ; R m ). As a consequence, the operator in (2.3) extends to a linearcontinuous operator from L p ( R n ; R m ) to L q ( R n ; R m ), for which we retain the same nota-tion. For a proof of (2.4) and (2.5), we refer the reader to [38, Chapter V, Section 1] andto [18, Section 1.2.1].We can now recall the following result, see [10, Proposition 2.2 and Corollary 2.3]. Proposition 2.1.
Let α ∈ (0 , . If f ∈ Lip c ( R n ) , then ∇ α f = I − α ∇ f = ∇ I − α f (2.6) and ∇ α f ∈ L ( R n ; R n ) ∩ L ∞ ( R n ; R n ) , with k∇ α f k L ( R n ; R n ) ≤ µ n,α [ f ] W α, ( R n ) (2.7) and k∇ α f k L ∞ ( R n ; R n ) ≤ C n,α,U k∇ f k L ∞ ( R n ; R n ) (2.8) for any bounded open set U ⊂ R n such that supp( f ) ⊂ U , where C n,α,U := nµ n,α (1 − α )( n + α − ω n diam( U ) − α + (cid:18) nω n n + α − (cid:19) n + α − n | U | − αn ! . (2.9) Analogously, if ϕ ∈ Lip c ( R n ; R n ) then div α ϕ = I − α div ϕ = div I − α ϕ (2.10) and div α ϕ ∈ L ( R n ) ∩ L ∞ ( R n ) , with k div α ϕ k L ( R n ) ≤ µ n,α [ ϕ ] W α, ( R n ; R n ) (2.11) and k div α ϕ k L ∞ ( R n ) ≤ C n,α,U k div ϕ k L ∞ ( R n ) (2.12) for any bounded open set U ⊂ R n such that supp( ϕ ) ⊂ U , where C n,α,U is as in (2.9) . Extension of ∇ α and div α to Lip b -regular tests. In the following result, weextend the fractional α -divergence to Lip b -regular vector fields. Lemma 2.2 (Extension of div α to Lip b ) . Let α ∈ (0 , . The operator div α : Lip b ( R n ; R n ) → L ∞ ( R n ) given by div α ϕ ( x ) := µ n,α Z R n ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 dy, x ∈ R n , (2.13) for all ϕ ∈ Lip b ( R n ; R n ) , is well defined, with k div α ϕ k L ∞ ( R n ) ≤ − α nω n µ n,α α (1 − α ) Lip( ϕ ) α k ϕ k − αL ∞ ( R n ; R n ) , (2.14) and satisfies div α ϕ ( x ) = µ n,α lim ε → + Z {| y − x | >ε } ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 dy = µ n,α lim ε → + Z {| y − x | >ε } ( y − x ) · ϕ ( y ) | y − x | n + α +1 dy (2.15) for all x ∈ R n . Moreover, if in addition I − α | div ϕ | ∈ L ( R n ) , then div α ϕ ( x ) = I − α div ϕ ( x ) (2.16) for a.e. x ∈ R n .Proof. We split the proof in two steps.
Step 1: proof of (2.13) , (2.14) and (2.15). Given x ∈ R n and r >
0, we can estimate Z {| y − x |≤ r } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy ≤ nω n Lip( ϕ ) Z r ̺ − α d̺ and Z {| y − x | >r } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy ≤ nω n k ϕ k L ∞ ( R n ; R n ) Z + ∞ r ̺ − (1+ α ) d̺. Hence the function in (2.13) is well defined for all x ∈ R n and k div α ϕ k L ∞ ( R n ) ≤ nω n Lip( ϕ )1 − α r − α + 2 k ϕ k L ∞ ( R n ; R n ) α r − α ! , so that (2.14) follows by optimising the right-hand side in r >
0. Moreover, since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y − x ) · ( ϕ ( y ) − ϕ ( x )) | y − x | n + α +1 χ ( ε, + ∞ ) ( | y − x | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Lip( ϕ ) χ (0 , ( | y − x | ) | y − x | n + α − + 2 k ϕ k L ∞ ( R n ; R n ) χ [1 , + ∞ ) ( | y − x | ) | y − x | n + α ∈ L x,y ( R n )and Z {| z | >ε } z | z | n + α +1 dy = 0for all ε >
0, by Lebesgue’s Dominated Convergence Theorem we immediately get thetwo equalities in (2.15) for all x ∈ R n . Step 2: proof of (2.16). Assume that I − α | div ϕ | ∈ L ( R n ). Then | div ϕ ( y ) || y − x | n + α − ∈ L y ( R n ) (2.17)for a.e. x ∈ R n . Hence, by Lebesgue’s Dominated Convergence Theorem, we can write I − α div ϕ ( x ) = µ n,α lim ε → + Z {| y − x | >ε } div ϕ ( y ) | y − x | n + α − dy DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 13 for a.e. x ∈ R n . Now let ε > R >
0. Again by (2.17) and Lebesgue’sDominated Convergence Theorem, we havelim R → + ∞ Z { R> | y − x | >ε } div ϕ ( y ) | y − x | n + α − dy = Z {| y − x | >ε } div ϕ ( y ) | y − x | n + α − dy for a.e. x ∈ R n . Moreover, integrating by parts, we get Z { R> | y − x | >ε } div ϕ ( y ) | y − x | n + α − dy = Z { R> | y | >ε } div y ϕ ( y + x ) | y | n + α − dy = Z {| y | = R } y | y | ϕ ( y + x ) | y | n + α − d H n − ( y ) − Z {| y | = ε } y | y | ϕ ( y + x ) | y | n + α − d H n − ( y )+ Z { R> | y | >ε } y · ϕ ( y + x ) | y | n + α +1 dy for all R > x ∈ R n . Since ϕ ∈ L ∞ ( R n ; R n ), by Lebesgue’s DominatedConvergence Theorem we havelim R → + ∞ Z { R> | y | >ε } y · ϕ ( y + x ) | y | n + α +1 dy = Z {| y | >ε } y · ϕ ( y + x ) | y | n + α +1 dy for all ε > x ∈ R n . We can also estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| y | = R } y | y | ϕ ( y + x ) | y | n + α − d H n − ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ nω n k ϕ k L ∞ ( R n ; R n ) R − α for all R > x ∈ R n . We thus have that Z {| y − x | >ε } div ϕ ( y ) | y − x | n + α − dy = Z {| y | >ε } y · ϕ ( y + x ) | y | n + α +1 dy − Z {| y | = ε } y | y | ϕ ( y + x ) | y | n + α − d H n − ( y )for all ε > x ∈ R n . Since also (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| y | = ε } y | y | ϕ ( y + x ) | y | n + α − d H n − ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| y | = ε } y | y | ϕ ( y + x ) − ϕ ( x ) | y | n + α − d H n − ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ nω n Lip( ϕ ) ε − α for all ε > x ∈ R n , we conclude thatlim ε → + Z {| y − x | >ε } div ϕ ( y ) | y − x | n + α − dy = lim ε → + Z {| y − x | >ε } ( y − x ) · ϕ ( y ) | y − x | n + α +1 dy for a.e. x ∈ R n , proving (2.16). (cid:3) We can also extend the fractional α -gradient to Lip b -regular functions. The proof isvery similar to the one of Lemma 2.2 and is left to the reader. Lemma 2.3 (Extension of ∇ α to Lip b ) . Let α ∈ (0 , . The operator ∇ α : Lip b ( R n ) → L ∞ ( R n ; R n ) given by ∇ α f ( x ) := µ n,α Z R n ( y − x ) · ( f ( y ) − f ( x )) | y − x | n + α +1 dy, x ∈ R n , for all f ∈ Lip b ( R n ) , is well defined, with k∇ α f k L ∞ ( R n ; R n ) ≤ − α nω n µ n,α α (1 − α ) Lip( f ) α k f k − αL ∞ ( R n ) , and satisfies ∇ α f ( x ) = µ n,α lim ε → + Z {| y − x | >ε } ( y − x ) · ( f ( y ) − f ( x )) | y − x | n + α +1 dy = µ n,α lim ε → + Z {| y − x | >ε } ( y − x ) · f ( y ) | y − x | n + α +1 dy for all x ∈ R n . Moreover, if in addition I − α |∇ f | ∈ L ( R n ) , then ∇ α f ( x ) = I − α ∇ f ( x ) for a.e. x ∈ R n . Extended Leibniz’s rules for ∇ α and div α . The following two results extend thevalidity of Leibniz’s rules proved in [10, Lemmas 2.6 and 2.7] to Lip b -regular functionsand Lip b -regular vector fields. The proofs are very similar to the ones given in [10] andto those of Lemma 2.2 and Lemma 2.3, and thus are left to the reader. Lemma 2.4 (Extended Leibniz’s rule for ∇ α ) . Let α ∈ (0 , . If f ∈ Lip b ( R n ) and η ∈ Lip c ( R n ) , then ∇ α ( ηf ) = η ∇ α f + f ∇ α η + ∇ α NL ( η, f ) , where ∇ α NL ( η, f )( x ) = µ n,α Z R n ( y − x ) · ( f ( y ) − f ( x ))( η ( y ) − η ( x )) | y − x | n + α +1 dy for all x ∈ R n , with k∇ α NL ( η, f ) k L ∞ ( R n ; R n ) ≤ − α nω n µ n,α k f k L ∞ ( R n ) α (1 − α ) Lip( η ) α k η k − αL ∞ ( R n ) and k∇ α NL ( η, f ) k L ( R n ; R n ) ≤ µ n,α k f k L ∞ ( R n ) [ η ] W α, ( R n ) . Lemma 2.5 (Extended Leibniz’s rule for div α ) . Let α ∈ (0 , . If ϕ ∈ Lip b ( R n ; R n ) and η ∈ Lip c ( R n ) , then div α ( ηϕ ) = η div α ϕ + ϕ · ∇ α η + div α NL ( η, ϕ ) , where div α NL ( η, ϕ )( x ) = µ n,α Z R n ( y − x ) · ( ϕ ( y ) − ϕ ( x ))( η ( y ) − η ( x )) | y − x | n + α +1 dy for all x ∈ R n , with k div α NL ( η, ϕ ) k L ∞ ( R n ) ≤ − α nω n µ n,α k ϕ k L ∞ ( R n ; R n ) α (1 − α ) Lip( η ) α k η k − αL ∞ ( R n ) and k div α NL ( η, ϕ ) k L ( R n ) ≤ µ n,α k ϕ k L ∞ ( R n ; R n ) [ η ] W α, ( R n ) . DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 15
Extended integration-by-part formulas.
We now recall the definition of thespace of functions with bounded fractional α -variation. Given α ∈ (0 , BV α ( R n ) := n f ∈ L ( R n ) : | D α f | ( R n ) < + ∞ o , where | D α f | ( R n ) = sup (cid:26)Z R n f div α ϕ dx : ϕ ∈ C ∞ c ( R n ; R n ) , k ϕ k L ∞ ( R n ; R n ) ≤ (cid:27) is the fractional α -variation of f ∈ L ( R n ). We refer the reader to [10, Section 3] for thebasic properties of this function space. Here we just recall the following result, see [10,Theorem 3.2 and Proposition 3.6] for the proof. Theorem 2.6 (Structure theorem for BV α functions) . Let α ∈ (0 , . If f ∈ L ( R n ) ,then f ∈ BV α ( R n ) if and only if there exists a finite vector-valued Radon measure D α f ∈ M ( R n ; R n ) such that Z R n f div α ϕ dx = − Z R n ϕ · dD α f (2.18) for all ϕ ∈ Lip c ( R n ; R n ) . Thanks to Lemma 2.5, we can actually prove that a function in BV α ( R n ) can be testedagainst any Lip b -regular vector field. Proposition 2.7 (Lip b -regular test for BV α functions) . Let α ∈ (0 , . If f ∈ BV α ( R n ) ,then (2.18) holds for all ϕ ∈ Lip b ( R n ; R n ) .Proof. We argue similarly as in the proof of [10, Theorem 3.8]. Fix ϕ ∈ Lip b ( R n ; R n ) andlet ( η R ) R> ⊂ C ∞ c ( R n ) be a family of cut-off functions as in [10, Section 3.3]. On the onehand, since (cid:12)(cid:12)(cid:12)(cid:12)Z R n f η R div α ϕ dx − Z R n f div α ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k div α ϕ k L ∞ ( R n ) Z R n | f | (1 − η R ) dx for all R >
0, by Lebesgue’s Dominated Convergence Theorem we havelim R → + ∞ Z R n f η R div α ϕ dx = Z R n f div α ϕ dx. On the other hand, by Lemma 2.5 we can write Z R n f η R div α ϕ dx = Z R n f div α ( η R ϕ ) dx − Z R n f ϕ · ∇ α η R dx − Z R n f div α NL ( η R , ϕ ) dx for all R >
0. By [10, Proposition 3.6], we have Z R n f div α ( η R ϕ ) dx = − Z R n η R ϕ · dD α f for all R >
0. Since (cid:12)(cid:12)(cid:12)(cid:12)Z R n η R ϕ · dD α f − Z R n ϕ · dD α f (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k L ∞ ( R n ; R n ) Z R n (1 − η R ) d | D α f | for all R >
0, by Lebesgue’s Dominated Convergence Theorem (with respect to the finitemeasure | D α f | ) we have lim R → + ∞ Z R n η R ϕ · dD α f = Z R n ϕ · dD α f. Finally, we can estimate (cid:12)(cid:12)(cid:12)(cid:12)Z R n f ϕ · ∇ α η R dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k L ∞ ( R n ; R n ) Z R n | f ( x ) | Z R n | η R ( y ) − η R ( x ) || y − x | n + α dy dx and, similarly, (cid:12)(cid:12)(cid:12)(cid:12)Z R n f div α NL ( η R , ϕ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k L ∞ ( R n ; R n ) Z R n | f ( x ) | Z R n | η R ( y ) − η R ( x ) || y − x | n + α dy dx. By Lebesgue’s Dominated Convergence Theorem, we thus get thatlim R → + ∞ (cid:18)Z R n f ϕ · ∇ α η R dx + Z R n f div α NL ( η R , ϕ ) dx (cid:19) = 0and the conclusion follows. (cid:3) Thanks to Lemma 2.4, we can prove that a function in Lip b ( R n ) can be tested againstany Lip c -regular vector field. The proof is very similar to the one of Proposition 2.7 andis thus left to the reader. Proposition 2.8 (Integration by parts for Lip b -regular functions) . Let α ∈ (0 , . If f ∈ Lip b ( R n ) , then Z R n f div α ϕ dx = − Z R n ϕ · ∇ α f dx for all ϕ ∈ Lip c ( R n ; R n ) . Comparison between W α, and BV α seminorms. In this section, we completelyanswer a question left open in [10, Section 1.4]. Given α ∈ (0 ,
1) and an open set Ω ⊂ R n ,we want to study the equality cases in the inequalities k∇ α f k L ( R n ; R n ) ≤ µ n,α [ f ] W α, ( R n ) , | D α χ E | (Ω) ≤ µ n,α P α ( E ; Ω) , as long as f ∈ W α, ( R n ) and P α ( E ; Ω) < + ∞ . The key idea to the solution of thisproblem lies in the following simple result. Lemma 2.9.
Let A ⊂ R n be a measurable set with L n ( A ) > . If F ∈ L ( A ; R m ) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z A F ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z A | F ( x ) | dx, with equality if and only if F = f ν a.e. in A for some constant direction ν ∈ S m − andsome scalar function f ∈ L ( A ) with f ≥ a.e. in A .Proof. The inequality is well known and it is obvious that it is an equality if F = f ν a.e. in A for some constant direction ν ∈ S m − and some scalar function f ∈ L ( A ) with f ≥ A . So let us assume that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z A F ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z A | F ( x ) | dx. If R A F ( x ) dx = 0, then also R A | F ( x ) | dx = 0. Thus F = 0 a.e. in A and there is nothingto prove. If R A F ( x ) dx = 0 instead, then we can write Z A | F ( x ) | − F ( x ) · ν dx = 0 , DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 17 with ν = R A F ( x ) dx | R A F ( x ) dx | ∈ S m − . Therefore, we obtain | F ( x ) | = F ( x ) · ν for a.e. x ∈ A , so that F ( x ) | F ( x ) | · ν = 1 for a.e. x ∈ A such that | F ( x ) | 6 = 0. This implies that F = f ν a.e. in A with f = | F | ∈ L ( A ) and theconclusion follows. (cid:3) As an immediate consequence of Lemma 2.9, we have the following result.
Corollary 2.10.
Let α ∈ (0 , . If f ∈ W α, ( R n ) , then k∇ α f k L ( R n ; R n ) ≤ µ n,α [ f ] W α, ( R n ) , (2.19) with equality if and only if f = 0 a.e. in R n .Proof. Inequality (2.19) was proved in [10, Theorem 3.18]. Note that, given f ∈ L ( R n ),[ f ] W α, ( R n ) = 0 if and only if f = 0 a.e. and thus, in this case, (2.19) is trivially an equality.If (2.19) holds as an equality and f is not equivalent to the zero function, then Z R n |∇ α f ( x ) | − µ n,α Z R n | f ( y ) − f ( x ) || y − x | n + α dy ! dx = 0and thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ( f ( y ) − f ( x )) · ( y − x ) | y − x | n + α +1 dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z R n | f ( y ) − f ( x ) || y − x | n + α dy (2.20)for all x ∈ U , for some measurable set U ⊂ R n such that L n ( R n \ U ) = 0. Now let x ∈ U be fixed. By Lemma 2.9 (applied with A = R n ), (2.20) implies that the (non-identicallyzero) vector field y ( f ( y ) − f ( x )) ( y − x ) , y ∈ R n , has constant direction for all y ∈ V x , for some measurable set V x ⊂ R n such that L n ( R n \ V x ) = 0. Thus, given y, y ′ ∈ V x , the two vectors y − x and y ′ − x are lin-early dependent, so that the three points x , y and y ′ are collinear. If n ≥
2, then thisimmediately gives L n ( V x ) = 0, a contradiction, so that (2.19) must be strict. If instead n = 1, then we know that x ∈ U = ⇒ y ( f ( y ) − f ( x )) ( y − x ) has constant sign for all y ∈ V x . (2.21)We claim that (2.21) implies that the function f is (equivalent to) a (non-constant)monotone function. If so, then f / ∈ L ( R ), in contrast with the fact that f ∈ W α, ( R ),so that (2.19) must be strict and the proof is concluded. To prove the claim, we argue asfollows. Fix x ∈ U and assume that( f ( y ) − f ( x )) ( y − x ) > y ∈ V x without loss of generality. Now pick x ′ ∈ U ∩ V x such that x ′ > x . Then,choosing y = x ′ in (2.22), we get ( f ( x ′ ) − f ( x )) ( x ′ − x ) > f ( x ′ ) > f ( x ).Similarly, if x ′ ∈ U ∩ V x is such that x ′ < x , then f ( x ′ ) < f ( x ). Henceess sup z
Let α ∈ (0 , , Ω ⊂ R n be an open set and E ⊂ R n be a measurable setsuch that ˜ P α ( E ; Ω) < + ∞ .(i) If n ≥ , L n ( E ) > and L n ( R n \ E ) > , then inequality (2.23) is strict.(ii) If n = 1 , then (2.23) is an equality if and only if the following hold:(a) for a.e. x ∈ Ω ∩ E , L (( −∞ , x ) \ E ) = 0 vel L (( x, + ∞ ) \ E ) = 0 ;(b) for a.e. x ∈ Ω \ E , L (( −∞ , x ) ∩ E ) = 0 vel L (( x, + ∞ ) ∩ E ) = 0 .Proof. We prove the two statements separately.
Proof of (i). Assume n ≥
2. Since L n ( E ) >
0, for a given x ∈ Ω \ E the map y ( y − x ) , for y ∈ E, does not have constant orientation. Similarly, since L n ( R n \ E ) >
0, for a given x ∈ Ω ∩ E also the map y ( y − x ) , for y ∈ R n \ E, does not have constant orientation. Hence, by Lemma 2.9, we must have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z E y − x | y − x | n + α +1 dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Z E dy | y − x | n + α , for x ∈ Ω \ E, and, similarly, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n \ E y − x | y − x | n + α +1 dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Z R n \ E dy | y − x | n + α , for x ∈ Ω ∩ E. We thus get k∇ α χ E k L (Ω; R n ) = µ n,α Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ( χ E ( y ) − χ E ( x )) · ( y − x ) | y − x | n + α +1 dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = µ n,α Z Ω \ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z E y − x | y − x | n + α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx + µ n,α Z Ω ∩ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n \ E y − x | y − x | n + α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx< µ n,α Z Ω \ E Z E dy dx | y − x | n + α + µ n,α Z Ω ∩ E Z R n \ E dy dx | y − x | n + α = µ n,α ˜ P α ( E ; Ω) , proving (i). DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 19
Proof of (ii). Assume n = 1. We argue as in the proof of [10, Proposition 4.12]. Let f E ( y, x ) := χ E ( y ) − χ E ( x ) | y − x | α , for x, y ∈ R , y = x. Then we can write˜ P α ( E ; Ω) = Z Ω Z R | f E ( y, x ) | dy dx = Z Ω Z x −∞ | f E ( y, x ) | dy + Z + ∞ x | f E ( y, x ) | dy ! dx and k∇ α χ E k L (Ω; R ) = µ ,α Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R f E ( y, x ) sgn( y − x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = µ ,α Z Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy − Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx. Hence (2.23) is an equality if and only if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy − Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z x −∞ | f E ( y, x ) | dy + Z + ∞ x | f E ( y, x ) | dy (2.24)for a.e. x ∈ Ω. Observing that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy − Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z x −∞ | f E ( y, x ) | dy + Z + ∞ x | f E ( y, x ) | dy for a.e. x ∈ Ω, we deduce that (2.23) is an equality if and only if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy − Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.25)= Z x −∞ | f E ( y, x ) | dy + Z + ∞ x | f E ( y, x ) | dy (2.26)for a.e. x ∈ Ω. Now, on the one hand, squaring both sides of (2.25) and simplifying, weget that (2.23) is an equality if and only if Z x −∞ f E ( y, x ) dy ! Z + ∞ x f E ( y, x ) dy ! = 0 (2.27)for a.e. x ∈ Ω. On the other hand, we can rewrite (2.26) as0 ≤ Z x −∞ | f E ( y, x ) | dy − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Z + ∞ x | f E ( y, x ) | dy ≤ x ∈ Ω, so that we must have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z x −∞ f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z x −∞ | f E ( y, x ) | dy and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z + ∞ x f E ( y, x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z + ∞ x | f E ( y, x ) | dy for a.e. x ∈ Ω. Hence (2.27) can be equivalently rewritten as Z x −∞ | f E ( y, x ) | dy ! Z + ∞ x | f E ( y, x ) | dy ! = 0 (2.28)for a.e. x ∈ Ω. Thus (2.23) is an equality if and only if at least one of the two integrals inthe left-hand side of (2.28) is zero, and the reader can check that (ii) readily follows. (cid:3)
Remark 2.12 (Half-lines in Corollary 2.11(ii)) . In the case n = 1, it is worth to stressthat (2.23) is always an equality when the set E ⊂ R is (equivalent to) an half-line, i.e., k∇ α χ ( a, + ∞ ) k L (Ω; R ) = µ ,α ˜ P α (( a, + ∞ ); Ω)for any α ∈ (0 , a ∈ R and any open set Ω ⊂ R such that ˜ P α (( a, + ∞ ); Ω) < + ∞ .However, the equality cases in (2.23) are considerably richer. Indeed, on the one side, k∇ α χ ( − , − ∪ ( − , + ∞ ) k L ((0 , R ) = µ ,α ˜ P α (( − , − ∪ ( − , + ∞ ); (0 , k∇ α χ ( − , − ∪ (0 , + ∞ ) k L (( − , R ) < µ ,α ˜ P α (( − , − ∪ (0 , + ∞ ); ( − , α ∈ (0 , Estimates and representation formulas for the fractional α -gradient Integrability properties of the fractional α -gradient. We begin with the fol-lowing technical local estimate on the W α, -seminorm of a function in BV loc . Lemma 3.1.
Let α ∈ (0 , and let f ∈ BV loc ( R n ) . Then f ∈ W α, ( R n ) with [ f ] W α, ( B R ) ≤ nω n (2 R ) − α − α | Df | ( B R ) (3.1) for all R > .Proof. Fix
R > f ∈ BV loc ( R n ) be such that f ∈ C ( B R ). We can estimate[ f ] W α, ( B R ) = Z B R Z B R | f ( y ) − f ( x ) || y − x | n + α dy dx = Z B R Z B R ∩{| y − x | < R } | f ( y ) − f ( x ) || y − x | n + α dy dx ≤ Z {| h | < R } | h | n + α Z B R | f ( x + h ) − f ( x ) | dx dh. Since Z B R | f ( x + h ) − f ( x ) | dx ≤ Z B R Z |∇ f ( x + th ) · h | dt dx ≤ | h | Z Z B R |∇ f ( x + th ) | dx dt ≤ | h | Z B R + | h | |∇ f ( z ) | dz DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 21 for all h ∈ R n , we have[ f ] W α, ( B R ) ≤ Z {| h | < R } | h | n + α − Z B R + | h | |∇ f ( z ) | dz dh ≤ Z {| h | < R } | Df | ( B R ) | h | n + α − dh = nω n (2 R ) − α − α | Df | ( B R )proving (3.1) for all f ∈ BV loc ( R n ) ∩ C ( B R ). Now fix R > f ∈ BV loc ( R n ).By [15, Theorem 5.3], there exists ( f k ) k ∈ N ⊂ BV ( B R ) ∩ C ∞ ( B R ) such that | Df k | ( B R ) →| Df | ( B R ) and f k → f a.e. in B R as k → + ∞ . Hence, by Fatou’s Lemma, we get[ f ] W α, ( B R ) ≤ lim inf k → + ∞ [ f k ] W α, ( B R ) ≤ nω n (2 R ) − α − α lim k → + ∞ | Df k | ( B R )= nω n (2 R ) − α − α | Df | ( B R )and the proof is complete. (cid:3) In the following result, we collect several local integrability estimates involving thefractional α -gradient of a function satisfying various regularity assumptions. Proposition 3.2.
The following statements hold.(i) If f ∈ BV ( R n ) , then f ∈ BV α ( R n ) for all α ∈ (0 , with D α f = ∇ α f L n and ∇ α f = I − α Df a.e. in R n . (3.2) In addition, for any bounded open set U ⊂ R n , we have k∇ α f k L ( U ; R n ) ≤ C n,α,U | Df | ( R n ) (3.3) for all α ∈ (0 , , where C n,α,U is as in (2.9) . Finally, given an open set A ⊂ R n ,we have k∇ α f k L ( A ; R n ) ≤ nω n µ n,α n + α − | Df | ( A r )1 − α r − α + n + 2 α − α k f k L ( R n ) r − α ! (3.4) for all r > and α ∈ (0 , , where A r := { x ∈ R n : dist( x, A ) < r } . In particular,we have k∇ α f k L ( R n ; R n ) ≤ nω n µ n,α ( n + 2 α − − α α (1 − α )( n + α − k f k − αL ( R n ) [ f ] αBV ( R n ) . (3.5) (ii) If f ∈ L ∞ ( R n ) ∩ W α, ( R n ) , then the weak fractional α -gradient D α f ∈ M loc ( R n ; R n ) exists and satisfies D α f = ∇ α f L n with ∇ α f ∈ L ( R n ; R n ) and k∇ α f k L ( B R ; R n ) ≤ µ n,α Z B R Z R n | f ( x ) − f ( y ) || x − y | n + α dx dy ≤ µ n,α (cid:16) [ f ] W α, ( B R ) + P α ( B R ) k f k L ∞ ( R n ) (cid:17) (3.6) for all R > and α ∈ (0 , . (iii) If f ∈ L ∞ ( R n ) ∩ BV loc ( R n ) , then the weak fractional α -gradient D α f ∈ M loc ( R n ; R n ) exists and satisfies D α f = ∇ α f L n with ∇ α f ∈ L ( R n ; R n ) and k∇ α f k L ( B R ; R n ) ≤ µ n,α nω n (2 R ) − α − α | Df | ( B R ) + 2( nω n ) R n − α α Γ(1 − α ) − k f k L ∞ ( R n ) ! . (3.7) for all R > and α ∈ (0 , .Proof. We prove the three statements separately.
Proof of (i) . Thanks to [10, Theorem 3.18], we just need to prove (3.3) and (3.4).We prove (3.3). By (3.2), by Tonelli’s Theorem and by [10, Lemma 2.4], we get Z U |∇ α f | dx ≤ Z U I − α | Df | dx ≤ C n,α,U | Df | ( R n ) , where C n,α,U is defined as in (2.9).We now prove (3.4) in two steps. Proof of (3.4) , Step 1 . Assume f ∈ C ∞ c ( R n ) and fix r >
0. We have Z A |∇ α f | dx = Z A | I − α ∇ f | dx ≤ µ n,α n + α − Z A Z {| h |≤ r } |∇ f ( x + h ) || h | n + α − dh dx + Z A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| h | >r } ∇ f ( x + h ) | h | n + α − dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ! . We estimate the two double integrals appearing in the right-hand side separately. ByTonelli’s Theorem, we have Z A Z {| h |≤ r } |∇ f ( x + h ) || h | n + α − dh dx = Z {| h |≤ r } Z A |∇ f ( x + h ) | dx dh | h | n + α − ≤ k∇ f k L ( A r ; R n ) Z {| h |≤ r } dh | h | n + α − = nω n r − α − α k∇ f k L ( A r ; R n ) . Concerning the second double integral, integrating by parts we get Z {| h | >r } ∇ f ( x + h ) | h | n + α − dh = ( n + α − Z {| h | >r } hf ( x + h ) | h | n + α +1 dh − Z {| h | = r } h | h | f ( x + h ) | h | n + α − d H n − ( h )for all x ∈ A . Hence, we can estimate Z A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| h | >r } ∇ f ( x + h ) | h | n + α − dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ ( n + α − Z A Z {| h | >r } | f ( x + h ) || h | n + α dh dx + Z A Z {| h | = r } | f ( x + h ) || h | n + α − d H n − ( h ) dx ≤ nω n k f k L ( R n ) r − α (cid:18) n + α − α + 1 (cid:19) = nω n (cid:18) n + 2 α − α (cid:19) k f k L ( R n ) r − α . DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 23
Thus (3.4) follows for all f ∈ C ∞ c ( R n ) and r > Proof of (3.4) , Step 2 . Let f ∈ BV ( R n ) and fix r >
0. Combining [15, Theorem 5.3]with a standard cut-off approximation argument, we find ( f k ) k ∈ N ⊂ C ∞ c ( R n ) such that f k → f in L ( R n ) and | Df k | ( R n ) → | Df | ( R n ) as k → + ∞ . By Step 1, we have that k∇ α f k k L ( A ; R n ) ≤ nω n µ n,α n + α − | Df k | ( A r )1 − α r − α + n + 2 α − α k f k k L ( R n ) r − α ! (3.8)for all k ∈ N . We claim that( ∇ α f k ) L n ⇀ ( ∇ α f ) L n as k → + ∞ . (3.9)Indeed, if ϕ ∈ Lip c ( R n ; R n ), then div α ϕ ∈ L ∞ ( R n ) by (2.12) and thus (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ α f k dx − Z R n ϕ · ∇ α f dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R n f k div α ϕ dx − Z R n f div α ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k div α ϕ k L ∞ ( R n ; R n ) k f k − f k L ( R n ) for all k ∈ N , so that lim k → + ∞ Z R n ϕ · ∇ α f k dx = Z R n ϕ · ∇ α f dx. Now fix ϕ ∈ C c ( R n ; R n ). Let U ⊂ R n be a bounded open set such that supp ϕ ⊂ U . Foreach ε > ψ ε ∈ Lip c ( R n ; R n ) such that k ϕ − ψ ε k L ∞ ( R n ; R n ) < ε andsupp ψ ε ⊂ U . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ϕ · ∇ α f k dx − Z R n ϕ · ∇ α f dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R n ψ ε · ∇ α f k dx − Z R n ψ ε · ∇ α f dx (cid:12)(cid:12)(cid:12)(cid:12) + k ψ ε − ϕ k L ∞ ( R n ; R n ) (cid:16) k∇ α f k k L ( U ; R n ) + k∇ α f k L ( U ; R n ) (cid:17) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R n ψ ε · ∇ α f k dx − Z R n ψ ε · ∇ α f dx (cid:12)(cid:12)(cid:12)(cid:12) + ε C n,α,U (cid:16) | Df k | ( R n ) + | Df | ( R n ) (cid:17) , so that lim k → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ α f k dx − Z R n ϕ · ∇ α f dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε C n,α,U | Df | ( R n ) . Thus, (3.9) follows passing to the limit as ε → + . Thanks to (3.9), by [22, Proposi-tion 4.29] we get that k∇ α f k L ( A ; R n ) ≤ lim inf k → + ∞ k∇ α f k k L ( A ; R n ) . Since | Df | ( U ) ≤ lim inf k → + ∞ | Df k | ( U )for any open set U ⊂ R n by [15, Theorem 5.2], we can estimatelim sup k → + ∞ | Df k | ( A r ) ≤ lim k → + ∞ | Df k | ( R n ) − lim inf k → + ∞ | Df k | ( R n \ A r ) ≤ | Df | ( R n ) − | Df | ( R n \ A r )= | Df | ( A r ) . Thus, (3.4) follows taking limits as k → + ∞ in (3.8). Finally, (3.5) is easily deduced byoptimising the right-hand side of (3.4) in the case A = R n with respect to r > Proof of (ii) . Assume f ∈ L ∞ ( R n ) ∩ W α, ( R n ). Given R >
0, we can estimate Z B R |∇ α f ( x ) | dx ≤ µ n,α Z B R Z R n | f ( x ) − f ( y ) || x − y | n + α dx dy = µ n,α Z B R Z B R | f ( x ) − f ( y ) || x − y | n + α dx dy + µ n,α Z B R Z R n \ B R | f ( x ) − f ( y ) || x − y | n + α dx dy ≤ µ n,α [ f ] W α, ( B R ) + 2 µ n,α k f k L ∞ ( R n ) Z B R Z R n \ B R | x − y | n + α dx dy = µ n,α [ f ] W α, ( B R ) + µ n,α k f k L ∞ ( R n ) P α ( B R )and (3.6) follows. To prove that D α f = ∇ α f L n , we argue as in the proof of [10, Propo-sition 4.8]. Let ϕ ∈ Lip c ( R n ; R n ). Since f ∈ L ∞ ( R n ), we have x
7→ | f ( x ) | Z R n | ϕ ( y ) − ϕ ( x ) || y − x | n + α dy ∈ L ( R n ) . Hence, by the definition of div α on Lip c -regular vector fields (see [10, Section 2.2]) and byLebesgue’s Dominated Convergence Theorem, we have Z R n f div α ϕ dx = lim ε → + Z R n f ( x ) Z {| y − x | >ε } ( y − x ) · ϕ ( y ) | y − x | n + α +1 dy dx. Since Z R n Z {| y − x | >ε } | f ( x ) | | ϕ ( y ) || y − x | n + α dy dx ≤ k f k L ∞ ( R n ) Z R n | ϕ ( y ) | Z {| y − x | >ε } | y − x | − n − α dx dy ≤ nω n αε α k f k L ∞ ( R n ) k ϕ k L ( R n ; R n ) for all ε >
0, by Fubini’s Theorem we can compute Z R n f ( x ) Z {| y − x | >ε } ( y − x ) · ϕ ( y ) | y − x | n + α +1 dy dx = − Z R n ϕ ( y ) Z {| x − y | >ε } ( x − y ) f ( x ) | x − y | n + α +1 dx dy = − Z R n ϕ ( y ) Z {| x − y | >ε } ( x − y ) ( f ( x ) − f ( y )) | x − y | n + α +1 dx dy. Since | ϕ ( y ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| x − y | >ε } ( x − y ) ( f ( x ) − f ( y )) | x − y | n + α +1 dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ϕ ( y ) | Z R n | f ( x ) − f ( y ) || x − y | n + α dx for all y ∈ R n and ε >
0, and y Z R n | f ( x ) − f ( y ) || x − y | n + α dx ∈ L ( R n )by (3.6), again by Lebesgue’s Dominated Convergence Theorem we conclude that Z R n f ( x ) div α ϕ ( x ) dx = − lim ε → Z R n ϕ ( y ) Z {| x − y | >ε } ( x − y ) ( f ( x ) − f ( y )) | x − y | n + α +1 dx dy = − Z R n ϕ ( y ) lim ε → Z {| x − y | >ε } ( x − y ) ( f ( x ) − f ( y )) | x − y | n + α +1 dx dy = − Z R n ϕ ( y ) · ∇ α f ( y ) dy DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 25 for all ϕ ∈ Lip c ( R n ; R n ). Thus D α f ∈ M loc ( R n ; R n ) is well defined and D α f = ∇ α f L n − . Proof of (iii) . Assume f ∈ L ∞ ( R n ) ∩ BV loc ( R n ). By Lemma 3.1, we know that f ∈ L ∞ ( R n ) ∩ W α, ( R n ) for all α ∈ (0 , D α f ∈ M loc ( R n ; R n ) exists by (ii). Hence,inserting (3.1) in (3.6), we find k∇ α f k L ( B R ; R n ) ≤ µ n,α nω n (2 R ) − α − α | Df | ( B R ) + P α ( B ) R n − α k f k L ∞ ( R n ) ! . Since for all x ∈ B we have Z R n \ B dy | y − x | n + α = Z R n \ B ( − x ) dz | z | n + α ≤ Z R n \ B −| x | dz | z | n + α = nω n α (1 − | x | ) α , being Γ increasing on (0 , + ∞ ) (see [4]), we can estimate P α ( B ) = 2 Z B Z R n \ B dy dx | y − x | n + α ≤ nω n α Z B dx (1 − | x | ) α = 2( nω n ) α Z t n − (1 − t ) α dt = 2( nω n ) α Γ( n ) Γ(1 − α )Γ( n + 1 − α ) ≤ nω n ) α Γ(1 − α ) , so that k∇ α f k L ( B R ; R n ) ≤ µ n,α nω n (2 R ) − α − α | Df | BV ( B R ) + 2( nω n ) R n − α α Γ(1 − α ) − k f k L ∞ ( R n ) ! , proving (3.7). (cid:3) Note that Proposition 3.2(i), in particular, applies to any f ∈ W , ( R n ). In the followingresult, we prove that a similar result holds also for any f ∈ W ,p ( R n ) with p ∈ (1 , + ∞ ). Proposition 3.3 ( W ,p ( R n ) ⊂ S α,p ( R n ) for p ∈ (1 , + ∞ )) . Let α ∈ (0 , and p ∈ (1 , + ∞ ) .If f ∈ W ,p ( R n ) , then f ∈ S α,p ( R n ) with k∇ αw f k L p ( A ; R n ) ≤ nω n µ n,α n + α − k∇ w f k L p ( A r ; R n ) − α r − α + n + 2 α − α k f k L p ( R n ) r − α ! (3.10) for any r > and any open set A ⊂ R n , where A r := { x ∈ R n : dist( x, A ) < r } . Inparticular, we have k∇ αw f k L p ( R n ; R n ) ≤ ( n + 2 α − − α n + α − nω n µ n,α α (1 − α ) k∇ w f k αL p ( R n ; R n ) k f k − αL p ( R n ) . (3.11) In addition, if p ∈ (cid:16) , n − α (cid:17) and q = npn − (1 − α ) p , then ∇ αw f = I − α ∇ w f a.e. in R n (3.12) and ∇ αw f ∈ L q ( R n ; R n ) .Proof. We argue similarly as in the proof of Proposition 3.2(i).
Proof of (3.10) , Step 1 . Assume f ∈ C ∞ c ( R n ) and fix an open set A ⊂ R n and r > I − α ∇ f ( x ) = µ n,α n + α − Z {| h |≤ r } ∇ f ( x + h ) | h | n + α − dh + Z {| h | >r } ∇ f ( x + h ) | h | n + α − dh ! = µ n,α n + α − Z {| h |≤ r } ∇ f ( x + h ) | h | n + α − dh + ( n + α − Z {| h | >r } h · f ( x + h ) | h | n + α +1 dh − Z {| h | = r } h | h | f ( x + h ) | h | n + α − d H n − ( h ) ! for all x ∈ A . By (2.6) and Minkowski’s Integral Inequality (see [38, Section A.1], forexample), we thus have k∇ α f k L p ( A ; R n ) ≤ µ n,α n + α − Z {| h |≤ r } k∇ f ( · + h ) k L p ( A ; R n ) | h | n + α − dh + ( n + α − Z {| h | >r } k f ( · + h ) k L p ( A ) | h | n + α dh + Z {| h | = r } k f ( · + h ) k L p ( A ) | h | n + α − d H n − ( h ) ! ≤ µ n,α n − α + 1 nω n − α k∇ f k L p ( A r ; R n ) r − α + nω n n + 2 α − α k f k L p ( R n ) r − α ! , proving (3.10) for all f ∈ C ∞ c ( R n ) and r > Proof of (3.10) , Step 2 . Let f ∈ W ,p ( R n ) and fix an open set A ⊂ R n and r > f k ) k ∈ N ⊂ C ∞ c ( R n ) such that f k → f in W ,p ( R n ) as k → + ∞ . By Step 1, we have that k∇ α f k k L p ( A ; R n ) ≤ nω n µ n,α n + α − k∇ f k k L p ( A r ; R n ) − α r − α + n + 2 α − α k f k k L p ( R n ) r − α ! (3.13)for all k ∈ N . Hence, choosing A = R n , we get that the sequence ( ∇ α f k ) k ∈ N is uniformlybounded in L p ( R n ; R n ). Up to pass to a subsequence (which we do not relabel for simplic-ity), there exists g ∈ L p ( R n ; R n ) such that ∇ α f k ⇀ g in L p ( R n ; R n ) as k → + ∞ . Given ϕ ∈ C ∞ c ( R n ; R n ), we have Z R n f k div α ϕ dx = − Z R n ϕ · ∇ α f k dx for all k ∈ N . Passing to the limit as k → + ∞ , by Proposition 2.1 we get that Z R n f div α ϕ dx = − Z R n ϕ · g dx for any ϕ ∈ C ∞ c ( R n ; R n ), so that g = ∇ αw f and hence f ∈ S α,p ( R n ) according to [10,Definition 3.19]. We thus have that k∇ αw f k L p ( A ; R n ) ≤ lim inf k → + ∞ k∇ α f k k L p ( A ; R n ) for any open set A ⊂ R n , since Z R n ϕ · ∇ αw f dx = lim k → + ∞ Z R n ϕ · ∇ α f k dx ≤ k ϕ k L pp − ( A ; R n ) lim inf k → + ∞ k∇ α f k k L p ( A ; R n ) for all ϕ ∈ C ∞ c ( A ; R n ). Therefore, (3.10) follows by taking limits as k → + ∞ in (3.13). Proof of (3.11). Inequality (3.11) follows by applying (3.10) with A = R n and minimis-ing the right-hand side with respect to r > DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 27
Proof of (3.12). Now assume p ∈ (cid:16) , n − α (cid:17) and let q = npn − (1 − α ) p . Let ϕ ∈ C ∞ c ( R n ; R n )be fixed. Recalling inequality (2.5), since ϕ ∈ L qq − ( R n ; R n ) we have that | ϕ | I − α | f | ∈ L ( R n ) , | ϕ | I − α |∇ w f | ∈ L ( R n ) . In particular, Fubini’s Theorem implies that f I − α ϕ ∈ L ( R n ; R n ) , I − α ϕ · ∇ w f ∈ L ( R n ) . Since div α ϕ ∈ L pp − ( R n ) by Proposition 2.1, we also get that f div I − α ϕ = f div α ϕ ∈ L ( R n ) . Therefore, observing that I − α ϕ ∈ Lip b ( R n ; R n ) because ∇ I − α ϕ = ∇ α ϕ ∈ L ∞ ( R n ; R n )again by Proposition 2.1 and performing a standard cut-off approximation argument, wecan integrate by parts and obtain Z R n ϕ · I − α ∇ w f dx = Z R n I − α ϕ · ∇ w f dx = − Z R n f div I − α ϕ dx = − Z R n f div α ϕ dx. Therefore Z R n ϕ · I − α ∇ w f dx = − Z R n f div α ϕ dx for all ϕ ∈ C ∞ c ( R n ; R n ), proving (3.12). In particular, notice that ∇ αw f ∈ L q ( R n ; R n ) byinequality (2.5). The proof is complete. (cid:3) For the case p = + ∞ , we have the following immediate consequence of Lemma 2.4 andProposition 2.8. Corollary 3.4 ( W , ∞ ( R n ) ⊂ S α, ∞ ( R n )) . Let α ∈ (0 , . If f ∈ W , ∞ ( R n ) , then f ∈ S α, ∞ ( R n ) with k∇ α f k L ∞ ( R n ; R n ) ≤ − α nω n µ n,α α (1 − α ) k∇ w f k αL ∞ ( R n ; R n ) k f k − αL ∞ ( R n ) . (3.14)3.2. Two representation formulas for the α -variation. In this section, we prove twouseful representation formulas for the α -variation.We begin with the following weak representation formula for the fractional α -variationof functions in BV loc ( R n ) ∩ L ∞ ( R n ). Here and in the following, we denote by f ⋆ the preciserepresentative of f ∈ L ( R n ), see (A.1) for the definition. Proposition 3.5.
Let α ∈ (0 , and f ∈ BV loc ( R n ) ∩ L ∞ ( R n ) . Then ∇ α f ∈ L ( R n ; R n ) and Z R n ϕ · ∇ α f dx = lim R → + ∞ Z R n ϕ · I − α ( χ ⋆B R Df ) dx (3.15) for all ϕ ∈ Lip c ( R n ; R n ) .Proof. By Proposition 3.2(iii), we know that ∇ α f ∈ L ( R n ; R n ) for all α ∈ (0 , f χ B R ∈ BV ( R n ) ∩ L ∞ ( R n ) with D ( χ B R f ) = χ ⋆B R Df + f ⋆ Dχ B R for all R >
0. Now fix ϕ ∈ Lip c ( R n ; R n ) and take R > ϕ ⊂ B R/ .By [10, Theorem 3.18], we have that Z R n χ B R f div α ϕ dx = − Z R n ϕ · ∇ α ( χ B R f ) dx = − Z R n ϕ · I − α D ( χ B R f ) dx. Moreover, we can split the last integral as Z R n ϕ · I − α D ( χ B R f ) dx = Z R n ϕ · I − α ( χ ⋆B R Df ) dx + Z R n ϕ · I − α ( f ⋆ Dχ B R ) dx. (3.16)For all x ∈ B R/ , we can estimate | I − α ( f ⋆ Dχ B R )( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂B R f ⋆ ( y ) | x − y | n + α − y | y | d H n − ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 R α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂B f ⋆ ( Ry ) (cid:12)(cid:12)(cid:12) y − xR (cid:12)(cid:12)(cid:12) n + α − y | y | d H n − ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ nω n R α (cid:16) − | x | R (cid:17) n + α − k f k L ∞ ( R n ) ≤ n + α − nω n R α k f k L ∞ ( R n ) and so, since supp ϕ ⊂ B R/ , we get that (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · I − α ( f ⋆ Dχ B R ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ n + α − nω n R α k ϕ k L ( R n ; R n ) k f k L ∞ ( R n ) . (3.17)Therefore, by (2.11), Lebesgue’s Dominated Convergence Theorem, (3.16) and (3.17), weget that Z R n f div α ϕ dx = lim R → + ∞ Z R n χ B R f div α ϕ dx = lim R → + ∞ Z R n ϕ · I − α ( χ ⋆B R Df ) dx and the conclusion follows. (cid:3) In the following result, we show that for all functions in bv ( R n ) ∩ L ∞ ( R n ) one canactually pass to the limit as R → + ∞ inside the integral in the right-hand side of (3.15). Corollary 3.6.
If either f ∈ BV ( R n ) or f ∈ bv ( R n ) ∩ L ∞ ( R n ) , then ∇ α f = I − α Df a.e. in R n . (3.18) Proof. If f ∈ BV ( R n ), then (3.18) coincides with (3.2) and there is nothing to prove. Solet us assume that f ∈ bv ( R n ) ∩ L ∞ ( R n ). Writing Df = ν f | Df | with ν f ∈ S n − | Df | -a.e.in R n , for all x ∈ R n we havelim R → + ∞ χ ⋆B R ( y ) ν f ( y ) | y − x | n + α − = ν f ( y ) | y − x | n + α − for | Df | -a.e. y = x. Moreover, for a.e. x ∈ R n , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ ⋆B R ( y ) ν f ( y ) | y − x | n + α − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | y − x | n + α − ∈ L y ( R n , | Df | ) ∀ R > , because I − α | Df | ∈ L ( R n ) by [10, Lemma 2.4]. Therefore, by Lebesgue’s DominatedConvergence Theorem (applied with respect to the finite measure | Df | ), we get thatlim R → + ∞ I − α ( χ ∗ B R Df )( x ) = ( I − α Df )( x ) for all x ∈ R n . Now let ϕ ∈ Lip c ( R n ; R n ). Since | ϕ · I − α ( χ ⋆B R Df ) | ≤ | ϕ | I − α | Df | ∈ L ( R n ) ∀ R > , DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 29 again by Lebesgue’s Dominated Convergence Theorem we get thatlim R → + ∞ Z R n ϕ · I − α ( χ ∗ B R Df ) dx = Z R n ϕ · I − α Df dx. (3.19)The conclusion thus follows combining (3.15) with (3.19). (cid:3)
Relation between BV β and BV α,p for β < α and p > . Let us recall thefollowing result, see [10, Lemma 3.28].
Lemma 3.7.
Let α ∈ (0 , . The following properties hold.(i) If f ∈ BV α ( R n ) , then u := I − α f ∈ bv ( R n ) with Du = D α f in M ( R n ; R n ) .(ii) If u ∈ BV ( R n ) , then f := ( − ∆) − α u ∈ BV α ( R n ) with k f k L ( R n ) ≤ c n,α k u k BV ( R n ) and D α f = Du in M ( R n ; R n ) . As a consequence, the operator ( − ∆) − α : BV ( R n ) → BV α ( R n ) is continuous. We can thus relate functions with bounded α -variation and functions with boundedvariation via Riesz potential and the fractional Laplacian. We would like to prove asimilar result between functions with bounded α -variation and functions with bounded β -variation, for any couple of exponents 0 < β < α < f ∈ L ( R n ) is well define, itis not clear whether the functional ϕ Z R n f div α ϕ dx (3.20)is well posed for all ϕ ∈ C ∞ c ( R n ; R n ), since div α ϕ does not have compact support. Nev-ertheless, thanks to Proposition 2.1, the functional in (3.20) is well defined as soon as f ∈ L p ( R n ) for some p ∈ [1 , + ∞ ]. Hence, it seems natural to define the space BV α,p ( R n ) := { f ∈ L p ( R n ) : | D α f | ( R n ) < ∞} (3.21)for any α ∈ (0 ,
1) and p ∈ [1 , + ∞ ]. In particular, BV α, ( R n ) = BV α ( R n ). Similarly, welet BV ,p ( R n ) := { f ∈ L p ( R n ) : | Df | ( R n ) < + ∞} for all p ∈ [1 , + ∞ ]. In particular, BV , ( R n ) = BV ( R n ).A further justification for the definition of these new spaces comes from the followingfractional version of the Gagliardo–Nirenberg–Sobolev embedding: if n ≥ α ∈ (0 , BV α ( R n ) is continuously embedded in L p ( R n ) for all p ∈ h , nn − α i , see [10,Theorem 3.9]. Hence, thanks to (3.21), we can equivalently write BV α ( R n ) ⊂ BV α,p ( R n )with continuous embedding for all n ≥ α ∈ (0 ,
1) and p ∈ h , nn − α i .Incidentally, we remark that the continuous embedding BV α ( R n ) ⊂ L nn − α ( R n ) for n ≥ α ∈ (0 ,
1) can be improved using the main result of the recent work [36] (see also [37]).Indeed, if n ≥ α ∈ (0 ,
1) and f ∈ C ∞ c ( R n ), then, by taking F = ∇ α f in [36, Theo-rem 1.1], we have that k f k L nn − α , ( R n ) ≤ c n,α k I α ∇ α f k L nn − α , ( R n ; R n ) ≤ c ′ n,α k∇ α f k L ( R n ; R n ) thanks to the boundedness of the Riesz transform R : L nn − α , ( R n ) → L nn − α , ( R n ; R n ), where c n,α , c ′ n,α > n and α , and L nn − α , ( R n ) is the Lorentzspace of exponents nn − α , BV α ( R n ) ⊂ L nn − α , ( R n ) for n ≥ α ∈ (0 ,
1) using Fatou’sLemma in Lorentz spaces (see [17, Exercise 1.4.11] for example). This suggests that thespaces defined in (3.21) may be further enlarged by considering functions belonging tosome Lorentz space, but we do not need this level of generality here.In the case n = 1, the space BV α ( R ) does not embed in L − α ( R ) with continuity,see [10, Remark 3.10]. However, somehow completing the picture provided by [36], wecan prove that the space BV α ( R ) continuously embeds in the Lorentz space L − α , ∞ ( R ).Although this result is truly interesting only for n = 1, we prove it below in all dimensionsfor the sake of completeness. Theorem 3.8 (Weak Gagliardo–Nirenberg–Sobolev inequality) . Let α ∈ (0 , . Thereexists a constant c n,α > such that k f k L nn − α , ∞ ( R n ) ≤ c n,α | D α f | ( R n ) (3.22) for all f ∈ BV α ( R n ) . As a consequence, BV α ( R n ) is continuously embedded in L q ( R n ) for any q ∈ [1 , nn − α ) .Proof. Let f ∈ C ∞ c ( R n ). By [35, Theorem 3.5] (see also [10, Section 3.6]), we have f ( x ) = − div − α ∇ α f ( x ) = − µ n, − α Z R n ( y − x ) · ∇ α f ( y ) | y − x | n +1 − α dy, x ∈ R n , so that | f ( x ) | ≤ µ n, − α Z R n |∇ α f ( y ) || y − x | n − α dy = µ n, − α µ n, − α ( n − α ) I α |∇ α f | ( x ) , x ∈ R n . Since I α : L ( R n ) → L nn − α , ∞ ( R n ) is a continuous operator by Hardy–Littlewood–Sobolevinequality (see [38, Theorem 1, Chapter V] or [17, Theorem 1.2.3]), we can estimate k f k L nn − α , ∞ ( R n ) ≤ n µ n, − α µ n, − α k I α |∇ α f |k L nn − α , ∞ ( R n ) ≤ c n,α k|∇ α f |k L ( R n ) = c n,α | D α f | ( R n ) , where c n,α > n and α . Thus, inequality (3.22) followsfor all f ∈ C ∞ c ( R n ). Now let f ∈ BV α ( R n ). By [10, Theorem 3.8], there exists ( f k ) k ∈ N ⊂ C ∞ c ( R n ) such that f k → f a.e. in R n and | D α f k | ( R n ) → | D α f | ( R n ) as k → + ∞ . ByFatou’s Lemma in Lorentz spaces (see [17, Exercise 1.4.11] for example), we thus get k f k L nn − α , ∞ ( R n ) ≤ lim inf k → + ∞ k f k k L nn − α , ∞ ( R n ) ≤ c n,α lim k → + ∞ | D α f k | ( R n ) = c n,α | D α f | ( R n )and so (3.22) readily follows. Finally, thanks to [17, Proposition 1.1.14], we obtain thecontinuous embedding of BV α ( R n ) in L q ( R n ) for all q ∈ [1 , nn − α ). (cid:3) Remark 3.9 (The embedding BV α ( R ) ⊂ L − α , ∞ ( R ) is sharp) . Let α ∈ (0 , BV α ( R ) ⊂ L − α , ∞ ( R ) is sharp at the level of Lorentz spaces, inthe sense that BV α ( R n ) \ L − α ,q ( R ) = ∅ for any q ∈ [1 , + ∞ ). Indeed, if we let f α ( x ) = | x − | α − sgn( x − − | x | α − sgn( x ) , x ∈ R \ { , } , DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 31 then f α ∈ BV α ( R ) by [10, Theorem 3.26], and it is not difficult to prove that f α ∈ L − α , ∞ ( R ). However, we can find a constant c α > | f α ( x ) | ≥ c α | x | α − χ ( − , )( x ) =: g α ( x ) , x ∈ R \ { , } , so that d f α ≥ d g α , where d f α and d g α are the distribution functions of f α and g α . A simplecalculation shows that d g α ( s ) =
12 if 0 < s ≤ c α − α (cid:18) c α t (cid:19) − α if s > c α − α , so that, by [17, Proposition 1.4.9], we obtain k f α k qL − α ,q ( R ) ≥ k g α k qL − α ,q ( R ) = 11 − α Z + ∞ [ d g α ( s )] q (1 − α ) s q − ds ≥ q (1 − α ) − α Z + ∞ c α − α s − q s q − ds = + ∞ and thus f α / ∈ L − α ,q ( R ) for any q ∈ [1 , + ∞ ).We collect the above continuous embeddings in the following statement. Corollary 3.10 (The embedding BV α ⊂ BV α,p ) . Let α ∈ (0 , and p ∈ h , nn − α (cid:17) . Wehave BV α ( R n ) ⊂ BV α,p ( R n ) with continuous embedding. In addition, if n ≥ , then also BV α ( R n ) ⊂ BV α, nn − α ( R n ) with continuous embedding. With Corollary 3.10 at hands, we are finally ready to investigate the relation between α -variation and β -variation for 0 < β < α < Lemma 3.11.
Let < β < α < . The following hold.(i) If f ∈ BV β ( R n ) , then u := I α − β f ∈ BV α,p ( R n ) for any p ∈ (cid:16) nn − α + β , nn − α (cid:17) (including p = nn − α if n ≥ ), with D α u = D β f in M ( R n ; R n ) .(ii) If u ∈ BV α ( R n ) , then f := ( − ∆) α − β u ∈ BV β ( R n ) with k f k L ( R n ) ≤ c n,α,β k u k BV α ( R n ) and D β f = D α u in M ( R n ; R n ) . As a consequence, the operator ( − ∆) α − β : BV α ( R n ) → BV β ( R n ) is continuous.Proof. We begin with the following observation. Let ϕ ∈ C ∞ c ( R n ; R n ) and let U ⊂ R n be a bounded open set such that supp ϕ ⊂ U . By Proposition 2.1 and the semigroupproperty (2.4) of the Riesz potential, we can writediv β ϕ = I − β div ϕ = I α − β I − α div ϕ = I α − β div α ϕ. Similarly, we also have I α − β | div α ϕ | = I α − β | I − α div ϕ | ≤ I α − β I − α | div ϕ | = I − β | div ϕ | , so that I α − β | div α ϕ | ∈ L ∞ ( R n ) with k I α − β | div α ϕ |k L ∞ ( R n ) ≤ k I − β | div ϕ |k L ∞ ( R n ) ≤ C n,β,U k div ϕ k L ∞ ( R n ) by [10, Lemma 2.4]. We now prove the two statements separately. Proof of (i) . Let f ∈ BV β ( R n ) and ϕ ∈ C ∞ c ( R n ; R n ). Thanks to Corollary 3.10, if n ≥ f ∈ BV β,q ( R n ) for any q ∈ [1 , nn − β ] and so I α − β f ∈ L p ( R n ) for any p ∈ (cid:16) nn − α + β , nn − α i by (2.5). If instead n = 1, then f ∈ BV β,q ( R ) for any q ∈ [1 , − β ) and so I α − β f ∈ L p ( R )for any p ∈ (cid:16) − α + β , − α (cid:17) . Since f ∈ L ( R n ) and I α − β | div α ϕ | ∈ L ∞ ( R n ), by Fubini’sTheorem we have Z R n f div β ϕ dx = Z R n f I α − β div α ϕ dx = Z R n u div α ϕ dx, (3.23)proving that u := I α − β f ∈ BV α,p ( R n ) for any p ∈ (cid:16) nn − α + β , nn − α (cid:17) (including p = nn − α if n ≥ D α u = D β f in M ( R n ; R n ). Proof of (ii) . Let u ∈ BV α ( R n ). By [10, Theorem 3.32], we know that u ∈ W α − β, ( R n ),so that f := ( − ∆) α − β u ∈ L ( R n ) with k f k L ( R n ) ≤ c n,α,β k u k BV α ( R n ) , see [10, Section 3.10].Then, arguing as before, for any ϕ ∈ C ∞ c ( R n ; R n ) we get (3.23), since we have I α − β f = u in L ( R n ) (see [10, Section 3.10]). The proof is complete. (cid:3) The inclusion BV α ⊂ W β, for β < α : a representation formula. In [10,Theorem 3.32], we proved that the inclusion BV α ⊂ W β, is continuous for β < α . In thefollowing result we prove a useful representation formula for the fractional β -gradient ofany f ∈ BV α ( R n ), extending the formula obtained in Corollary 3.6. Proposition 3.12.
Let α ∈ (0 , . If f ∈ BV α ( R n ) , then f ∈ W β, ( R n ) for all β ∈ (0 , α ) with ∇ β f = I α − β D α f a.e. in R n . (3.24) In addition, for any bounded open set U ⊂ R n , we have k∇ β f k L ( U ; R n ) ≤ C n, (1 − α + β ) ,U | D α f | ( R n ) (3.25) for all β ∈ (0 , α ) , where C n,α,U is as in (2.9) . Finally, given an open set A ⊂ R n , we have k∇ β f k L ( A ; R n ) ≤ µ n, α − β n + β − α ω n, | D α f | ( A r ) α − β r α − β + ω n,α ( n + 2 β − α ) β k f k L ( R n ) r − β ! (3.26) for all r > and all β ∈ (0 , α ) , where ω n,α := k∇ α χ B k L ( R n ; R n ) , ω n, := | Dχ B | ( R n ) = nω n , and, as above, A r := { x ∈ R n : dist( x, A ) < r } . In particular, we have k∇ β f k L ( R n ; R n ) ≤ αµ n, α − β ω βα n, ω − βα n,α ( n + 2 β − α ) − βα β ( n + β − α )( α − β ) k f k − βα L ( R n ) | D α f | ( R n ) βα . (3.27) Proof.
Fix β ∈ (0 , α ). By [10, Theorem 3.32] we already know that f ∈ W β, ( R n ), with D β f = ∇ β f L n according to [10, Theorem 3.18]. We thus just need to prove (3.24), (3.25)and (3.26).We prove (3.24). Let ϕ ∈ C ∞ c ( R n ; R n ). Note that I α − β ϕ ∈ Lip b ( R n ; R n ) is such thatdiv I α − β ϕ = I α − β div ϕ , so that I − α div I α − β ϕ = I − α I α − β div ϕ = I − β div ϕ = div β ϕ by the semigroup property (2.4) of the Riesz potential. Moreover, in a similar way, wehave I − α | div I α − β ϕ | = I − α | I α − β div ϕ | ≤ I − α I α − β | div ϕ | = I − β | div ϕ | ∈ L ( R n ) . DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 33
By Lemma 2.2, we thus have that div α I α − β ϕ = div β ϕ . Consequently, by Proposition 2.7,we get Z R n f div β ϕ dx = Z R n f div α I α − β ϕ dx = − Z R n I α − β ϕ · dD α f. Since | D α f | ( R n ) < + ∞ , we have I α − β | D α f | ∈ L ( R n ) and thus, by Fubini’s Theorem,we get that Z R n I α − β ϕ · dD α f = Z R n ϕ · I α − β D α f dx. We conclude that Z R n f div β ϕ dx = − Z R n ϕ · I α − β D α f dx for any ϕ ∈ C ∞ c ( R n ; R n ), proving (3.24).We prove (3.25). By (3.24), by Tonelli’s Theorem and by [10, Lemma 2.4], we get Z U |∇ β f | dx ≤ Z U I α − β | D α f | dx ≤ C n, (1 − α + β ) ,U | D α f | ( R n )where C n,α,U is as in (2.9).We now prove (3.26) in two steps. We argue similarly as in the proof of (3.4). Proof of (3.26) , Step 1 . Assume f ∈ C ∞ c ( R n ) and fix r >
0. We have Z A |∇ β f | dx = Z A | I α − β ∇ α f | dx ≤ µ n, β − α n + β − α Z A Z {| h |
0. By [10, Theorem 3.8], wefind ( f k ) k ∈ N ⊂ C ∞ c ( R n ) such that f k → f in L ( R n ) and | D α f k | ( R n ) → | D α f | ( R n ) as k → + ∞ . By Step 1, we have that k∇ β f k k L ( A ; R n ) ≤ µ n, β − α n + β − α nω n | D α f k | ( A r ) α − β r α − β + ω n,α ( n + 2 β − α ) β k f k k L ( R n ) r − β ! (3.28)for all k ∈ N . We have that( ∇ β f k ) L n ⇀ ( ∇ β f ) L n as k → + ∞ . (3.29)This can be proved arguing similarly as in the proof of (3.9) using (3.25). We leave thedetails to the reader. Thanks to (3.29), by [22, Proposition 4.29] we get that k∇ β f k L ( A ; R n ) ≤ lim inf k → + ∞ k∇ β f k k L ( A ; R n ) . Since | D α f | ( U ) ≤ lim inf k → + ∞ | D α f k | ( U )for any open set U ⊂ R n by [10, Theorem 3.3], we can estimatelim sup k → + ∞ | D α f k | ( A r ) ≤ lim k → + ∞ | D α f k | ( R n ) − lim inf k → + ∞ | D α f k | ( R n \ A r ) ≤ | D α f | ( R n ) − | D α f | ( R n \ A r )= | D α f | ( A r ) . Thus, (3.26) follows taking limits as k → + ∞ in (3.28). Finally, (3.27) follows by consid-ering A = R n in (3.26) and optimising the right-hand side in r > (cid:3) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 35 Asymptotic behaviour of fractional α -variation as α → − Convergence of ∇ α and div α as α → − . We begin with the following simpleresult about the asymptotic behaviour of the constant µ n,α as α → − . Lemma 4.1.
Let n ∈ N . We have µ n,α − α ≤ π − n s
32 Γ (cid:16) n + 1 (cid:17) Γ (cid:16) (cid:17) =: C n ∀ α ∈ (0 ,
1) (4.1) and lim α → − µ n,α − α = ω − n . (4.2) Proof.
Since Γ(1) = 1 and Γ(1 + x ) = x Γ( x ) for x > x ) ∼ x − as x → + . Thus as α → − we find µ n,α = 2 α π − n Γ (cid:16) n + α +12 (cid:17) Γ (cid:16) − α (cid:17) ∼ π − n (1 − α ) Γ (cid:18) n (cid:19) = ω − n (1 − α )and (4.2) follows.Since Γ is log-convex on (0 , + ∞ ) (see [4]), for all x > a ∈ (0 ,
1) we haveΓ( x + a ) = Γ((1 − a ) x + a ( x + 1)) ≤ Γ( x ) − a Γ( x + 1) a = x a Γ( x ) . For x = n and a = α +12 , we can estimateΓ (cid:18) n + α + 12 (cid:19) ≤ (cid:18) n (cid:19) α +12 Γ (cid:18) n (cid:19) ≤ Γ (cid:18) n (cid:19) for all n ≥
2. Also, for n = 1, we trivially have Γ (cid:16) α (cid:17) ≤ Γ (cid:16) (cid:17) , because Γ is increasingon (1 , + ∞ ) (see [4]). For x = 1 + − α and a = α , we can estimateΓ (cid:18) (cid:19) ≤ (cid:18) − α (cid:19) α Γ (cid:18) − α (cid:19) ≤ s
32 1 − α (cid:18) − α (cid:19) . We thus get µ n,α (1 − α ) − = 2 α − π − n Γ (cid:16) n + α +12 (cid:17) Γ (cid:16) − α + 1 (cid:17) ≤ π − n s
32 Γ (cid:16) n + 1 (cid:17) Γ (cid:16) (cid:17) and (4.1) follows. (cid:3) In the following technical result, we show that the constant C n,α,U defined in (2.9) isuniformly bounded as α → − in terms of the volume and the diameter of the boundedopen set U ⊂ R n . Lemma 4.2 (Uniform upper bound on C n,α,U as α → − ) . Let n ∈ N and α ∈ ( , . Let U ⊂ R n be bounded open set. If C n,α,U is as in (2.9) , then C n,α,U ≤ nω n C n (cid:16) n − (cid:17) n (cid:16) n − (cid:17) max ( , | U | ω n ) n + max (cid:26) , q diam( U ) (cid:27) =: κ n,U , (4.3) where C n is as in (4.1) . Proof.
By (4.1), for all α ∈ ( ,
1) we have n µ n,α ( n + α − − α ) ≤ n C n n + α − ≤ n C n n − . Since t − α ≤ max n , √ t o for any t ≥ α ∈ ( , ω n (diam( U )) − α ≤ ω n max (cid:26) , q diam( U ) (cid:27) and (cid:18) nω n n + α − (cid:19) n + α − n | U | − αn = nω n n + α − | U | ( n + α − nω n ! − αn ≤ nω n (cid:16) n − (cid:17) max ( , | U | ω n ) n . Combining these inequalities, we get the conclusion. (cid:3)
As consequence of Proposition 2.1 and Lemma 4.2, we prove that ∇ α and div α convergepointwise to ∇ and div respectively as α → − . Proposition 4.3. If f ∈ C c ( R n ) , then for all x ∈ R n we have lim α → − I α f ( x ) = f ( x ) . (4.4) As a consequence, if f ∈ C c ( R n ) and ϕ ∈ C c ( R n ; R n ) , then for all x ∈ R n we have lim α → − ∇ α f ( x ) = ∇ f ( x ) , lim α → − div α ϕ ( x ) = div ϕ ( x ) . (4.5) Proof.
Let f ∈ C c ( R n ) and fix x ∈ R n . Writing (2.6) in spherical coordinates, we find I α f ( x ) = µ n, − α n − α lim δ → Z ∂B Z + ∞ δ ̺ − α f ( x + ̺v ) d̺ d H n − ( v ) . Since f ∈ C c ( R n ), for each fixed v ∈ ∂B we can integrate by parts in the variable ̺ andget Z + ∞ δ ̺ − α f ( x + ̺v ) d̺ = (cid:20) ̺ α α f ( x + ̺v ) (cid:21) ̺ → + ∞ ̺ = δ − α Z + ∞ δ ̺ α ∂ ̺ ( f ( x + ̺v )) d̺ = − δ α α f ( x + δv ) − α Z + ∞ δ ̺ α ∂ ̺ ( f ( x + ̺v )) d̺. Clearly, we have lim δ → + δ α Z ∂B f ( x + δv ) d H n − ( v ) = 0 . Thus, by Fubini’s Theorem, we conclude that I α f ( x ) = − µ n, − α α ( n − α ) Z ∞ Z ∂B ̺ α ∂ ̺ ( f ( x + ̺v )) d H n − ( v ) d̺. (4.6)Since f has compact support and recalling (4.2), we can pass to the limit in (4.6) and getlim α → + I α f ( x ) = − nω n Z ∂B Z ∞ ∂ ̺ ( f ( x + ̺v )) d̺ d H n − ( v ) = f ( x ) , proving (4.4). The pointwise limits in (4.5) immediately follows by Proposition 2.1. (cid:3) In the following crucial result, we improve the pointwise convergence obtained in Propo-sition 4.3 to strong convergence in L p ( R n ) for all p ∈ [1 , + ∞ ]. DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 37
Proposition 4.4.
Let p ∈ [1 , + ∞ ] . If f ∈ C c ( R n ) and ϕ ∈ C c ( R n ; R n ) , then lim α → − k∇ α f − ∇ f k L p ( R n ; R n ) = 0 , lim α → − k div α ϕ − div ϕ k L p ( R n ) = 0 . Proof.
Let f ∈ C c ( R n ). Since Z B dy | y | n + α − = nω n Z d̺̺ α = nω n − α , for all x ∈ R n we can write nω n µ n,α (1 − α )( n + α − ∇ f ( x ) = µ n,α n + α − Z B ∇ f ( x ) | y | n + α − dy. Therefore, by (2.6), we have ∇ α f ( x ) − nω n µ n,α (1 − α )( n + α − ∇ f ( x )= µ n,α n + α − Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy + Z R n \ B ∇ f ( x + y ) | y | n + α − dy ! for all x ∈ R n . We now distinguish two cases. Case 1: p ∈ [1 , + ∞ ). Using the elementary inequality | v + w | p ≤ p − ( | v | p + | w | p ) validfor all v, w ∈ R n , we have Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ α f ( x ) − nω n µ n,α (1 − α )( n + α − ∇ f ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx ≤ p − µ n,α n + α − Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx + 2 p − µ n,α n + α − Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n \ B ∇ f ( x + y ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx. We now estimate the two double integrals appearing in the right-hand side separately.For the first double integral, similarly as in the proof of Proposition 4.3, we pass inspherical coordinates to get Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy = Z ∂B Z ̺ − α ( ∇ f ( x + ̺v ) − ∇ f ( x )) d̺ d H n − ( v )= 11 − α Z ∂B ( ∇ f ( x + v ) − ∇ f ( x )) d H n − ( v ) − Z ∂B Z ̺ − α − α ∂ ̺ ( ∇ f ( x + ̺v )) d̺ d H n − ( v ) (4.7)for all x ∈ R n . Hence, by (4.2), we findlim α → − µ n,α (1 − α )( n + α − Z ∂B ( ∇ f ( x + v ) − ∇ f ( x )) d H n − ( v )= 1 nω n Z ∂B ( ∇ f ( x + v ) − ∇ f ( x )) d H n − ( v )and lim α → − µ n,α (1 − α )( n + α − Z ∂B Z ̺ − α ∂ ̺ ( ∇ f ( x + ̺v )) d̺ d H n − ( v ) = 1 nω n Z ∂B Z ∂ ̺ ( ∇ f ( x + ̺v )) d̺ d H n − ( v )= 1 nω n Z ∂B ( ∇ f ( x + v ) − ∇ f ( x )) d H n − ( v )for all x ∈ R n . Therefore, we getlim α → − µ n,α n + α − Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy = 0for all x ∈ R n . Recalling (4.1), we also observe that µ n,α n + α − |∇ f ( x + y ) − ∇ f ( x ) || y | n + α − ≤ C n |∇ f ( x + y ) − ∇ f ( x ) || y | n for all α ∈ (0 , x ∈ R n and y ∈ B . Moreover, letting R > f ⊂ B R ,we can estimate Z B |∇ f ( x + y ) − ∇ f ( x ) || y | n dy ≤ nω n k∇ f k L ∞ ( R n ; R n ) χ B R +1 ( x )for all x ∈ R n , so that x Z B |∇ f ( x + y ) − ∇ f ( x ) || y | n dy ! p ∈ L ( R n ) . In conclusion, applying Lebesgue’s Dominated Convergence Theorem, we findlim α → − µ n,α n + α − Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx = 0 . For the second double integral, note that Z R n \ B ∇ f ( x + y ) | y | n + α − dy = Z R n \ B ∇ ( f ( x + y ) − f ( x )) | y | n + α − dy for all x ∈ R n . Now let R >
0. Integrating by parts, we have that Z B R \ B ∇ ( f ( x + y ) − f ( x )) | y | n + α − dy = ( n + α − Z B R \ B y ( f ( x + y ) − f ( x )) | y | n + α +1 dy + 1 R n + α − Z ∂B R ( f ( x + y ) − f ( x )) d H n − ( y ) − Z ∂B ( f ( x + y ) − f ( x )) d H n − ( y )for all x ∈ R n . Since Z R n \ B R | f ( x + y ) − f ( x ) || y | n + α dy ≤ nω n αR α k f k L ∞ ( R n ) and 1 R n + α − Z ∂B R | f ( x + y ) − f ( x ) | d H n − ( y ) ≤ nω n R α k f k L ∞ ( R n ) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 39 for all
R >
0, we conclude that Z R n \ B ∇ f ( x + y ) | y | n + α − dy = lim R → + ∞ Z B R \ B ∇ f ( x + y ) | y | n + α − dy = ( n + α − Z R n \ B y ( f ( x + y ) − f ( x )) | y | n + α +1 dy − Z ∂B ( f ( x + y ) − f ( x )) d H n − ( y ) (4.8)for all x ∈ R n . Hence, by Minkowski’s Integral Inequality (see [38, Section A.1], forexample), we can estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z R n \ B ∇ f ( · + y ) | y | n + α − dy (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ; R n ) ≤ ( n + α − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z R n \ B | f ( · + y ) − f ( · ) || y | n + α dy (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z ∂B | f ( · + y ) − f ( · ) | d H n − ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) ≤ n + 2 α − α nω n k f k L p ( R n ) . Thus, by (4.2), we get thatlim α → − µ n,α n + α − Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n \ B ∇ f ( x + y ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dx = 0 . Case 2: p = + ∞ . We havesup x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ α f ( x ) − nω n µ n,α (1 − α )( n + α − ∇ f ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ µ n,α n + α − sup x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n \ B ∇ f ( x + y ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! . Again we estimate the two integrals appearing in the right-hand side separately. We notethat Z ∂B ( ∇ f ( x + v ) − ∇ f ( x )) d H n − ( v ) − Z ∂B Z ̺ − α ∂ ̺ ( ∇ f ( x + ̺v )) d̺ d H n − ( v )= Z ∂B Z (1 − ̺ − α ) ∂ ̺ ( ∇ f ( x + ̺v )) d̺ d H n − ( v ) , so that we can rewrite (4.7) as Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy = 11 − α Z ∂B Z (1 − ̺ − α ) ∂ ̺ ( ∇ f ( x + ̺v )) d̺ d H n − ( v ) . Hence, we can estimatesup x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − α Z ∂B Z (1 − ̺ − α ) sup x ∈ R n | ∂ ̺ ( ∇ f ( x + ̺v )) | d̺ d H n − ( v ) ≤ − α nω n k∇ f k L ∞ ( R n ; R n ) , so that lim α → − µ n,α n + α − x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ∇ f ( x + y ) − ∇ f ( x ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . For the second integral, by (4.8) we can estimatesup x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n \ B ∇ f ( x + y ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ ( n + α −
1) sup x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n \ B | f ( x + y ) − f ( x ) || y | n + α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ∂B | f ( x + y ) − f ( x ) | d H n − ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n + 2 α − α nω n k f k L ∞ ( R n ) . Thus, by (4.2), we get thatlim α → − µ n,α n + α − x ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n \ B ∇ f ( x + y ) | y | n + α − dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . We can now conclude the proof. Again recalling (4.2), we thus find thatlim α → − k∇ α f − ∇ f k L p ( R n ; R n ) ≤ lim α → − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∇ α f − nω n µ n,α (1 − α )( n + α − ∇ f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ; R n ) + k∇ f k L p ( R n ; R n ) lim α → − nω n µ n,α (1 − α )( n + α − − ! = 0for all p ∈ [1 , + ∞ ] and the conclusion follows. The L p -convergence of div α ϕ to div ϕ as α → − for all p ∈ [1 , + ∞ ] follows by a similar argument and is left to the reader. (cid:3) Remark 4.5.
Note that the conclusion of Proposition 4.4 still holds if instead one assumesthat f ∈ S ( R n ) and ϕ ∈ S ( R n ; R n ), where S ( R n ; R m ) is the space of m -vector-valuedSchwartz functions. We leave the proof of this assertion to the reader.4.2. Weak convergence of α -variation as α → − . In Theorem 4.7 below, we provethat the fractional α -variation weakly converges to the standard variation as α → − forfunctions either in BV ( R n ) or in BV loc ( R n ) ∩ L ∞ ( R n ). In the proof of Theorem 4.7, weare going to use the following technical result. Lemma 4.6.
There exists a dimensional constant c n > with the following property. If f ∈ L ∞ ( R n ) ∩ BV loc ( R n ) , then k∇ α f k L ( B R ; R n ) ≤ c n (cid:16) R − α | Df | ( B R ) + R n − α k f k L ∞ ( R n ) (cid:17) (4.9) for all R > and α ∈ ( , .Proof. Since Γ( x ) ∼ x − as x → + (see [4]), inequality (4.9) follows immediately com-bining (3.7) with Lemma 4.1. (cid:3) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 41
Theorem 4.7.
If either f ∈ BV ( R n ) or f ∈ BV loc ( R n ) ∩ L ∞ ( R n ) , then D α f ⇀ Df as α → − . Proof.
We divide the proof in two steps.
Step 1 . Assume f ∈ BV ( R n ). By [10, Theorem 3.18], we have Z R n ϕ · ∇ α f dx = − Z R n f div α ϕ dx for all ϕ ∈ Lip c ( R n ; R n ). Thus, given ϕ ∈ C c ( R n ; R n ), recalling Proposition 4.3 and theestimates (2.12) and (4.3), by Lebesgue’s Dominated Convergence Theorem we get thatlim α → − Z R n ϕ · ∇ α f dx = − lim α → − Z R n f div α ϕ dx = − Z R n f div ϕ dx = Z R n ϕ · dDf. Now fix ϕ ∈ C c ( R n ; R n ). Let U ⊂ R n be a fixed bounded open set such that supp ϕ ⊂ U .For each ε > ψ ε ∈ C c ( R n ; R n ) such that k ϕ − ψ ε k L ∞ ( R n ; R n ) < ε and supp ψ ε ⊂ U . Then, by (3.3), we can estimate (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ α f dx − Z R n ϕ · dDf (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ − ψ ε k L ∞ ( R n ; R n ) (cid:18)Z U |∇ α f | dx + | Df | ( R n ) (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12)Z R n ψ ε · ∇ α f dx − Z R n ψ ε · dDf (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (1 + C n,α,U ) | Df | ( R n )+ (cid:12)(cid:12)(cid:12)(cid:12)Z R n ψ ε · ∇ α f dx − Z R n ψ ε · dDf (cid:12)(cid:12)(cid:12)(cid:12) for all α ∈ (0 , α → − (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ α f dx − Z R n ϕ · dDf (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (1 + κ n,U ) | Df | ( R n ) (4.10)and the conclusion follows passing to the limit as ε → + . Step 2 . Assume f ∈ BV loc ( R n ) ∩ L ∞ ( R n ). By Proposition 3.2(iii), we know that D α f = ∇ α f L n with ∇ α f ∈ L ( R n ; R n ). By Proposition 4.4, we get thatlim α → − (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ α f dx − Z R n ϕ · dDf (cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k L ∞ ( R n ) lim α → − k div α ϕ − div ϕ k L ( R n ; R n ) = 0for all ϕ ∈ C c ( R n ; R n ). Now fix ϕ ∈ C c ( R n ; R n ) and choose R ≥ ϕ ⊂ B R .For each ε > ψ ε ∈ C c ( R n ; R n ) such that k ϕ − ψ ε k L ∞ ( R n ; R n ) < ε and supp ψ ε ⊂ B R . Then, by (4.9), we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ϕ · ∇ α f dx − Z R n ϕ · dDf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ − ψ ε k L ∞ ( R n ; R n ) (cid:16) k∇ α f k L ( B R ; R n ) + | Df | ( B R ) (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ψ ε · ∇ α f dx − Z R n ψ ε · dDf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ εc n R n (cid:16) k f k L ∞ ( R n ) + | Df | ( B R ) (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ψ ε · ∇ α f dx − Z R n ψ ε · dDf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for all α ∈ ( , α → − (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ α f dx − Z R n ϕ · dDf (cid:12)(cid:12)(cid:12)(cid:12) ≤ εc n R n (cid:16) k f k L ∞ ( R n ) + | Df | ( B R ) (cid:17) (4.11)and the conclusion follows passing to the limit as ε → + . (cid:3) We are now going to improve the weak convergence of the fractional α -variation ob-tained in Theorem 4.7 by establishing the weak convergence also of the total fractional α -variation as α → − , see Theorem 4.9 below. To do so, we need the following prelimi-nary result. Lemma 4.8.
Let µ ∈ M ( R n ; R n ) . We have ( I α µ ) L n ⇀ µ as α → + .Proof. Since Riesz potential is a linear operator and thanks to Hahn–Banach Decompo-sition Theorem, without loss of generality we can assume that µ is a nonnegative finiteRadon measure.Let now ϕ ∈ C c ( R n ) and let U ⊂ R n be a bounded open set such that supp ϕ ⊂ U .We have that k I α | ϕ |k L ∞ ( R n ) ≤ κ n,U k ϕ k L ∞ ( R n ) for all α ∈ (0 , ) by [10, Lemma 2.4] andLemma 4.2. Thus, by (4.4), Fubini’s Theorem and Lebesgue’s Dominated ConvergenceTheorem, we get thatlim α → + Z R n ϕ I α µ dx = lim α → + Z R n I α ϕ dµ = Z R n ϕ dµ. Now fix ϕ ∈ C c ( R n ; R n ). Let U ⊂ R n be a fixed bounded open set such that supp ϕ ⊂ U .For each ε > ψ ε ∈ C c ( R n ; R n ) such that k ϕ − ψ ε k L ∞ ( R n ; R n ) < ε and supp ψ ε ⊂ U . Then, since µ ( R n ) < + ∞ , by [10, Lemma 2.4] and by (4.3), we canestimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ϕ I α µ dx − Z R n ϕ dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ψ ε I α µ dx − Z R n ψ ε dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ε k I α µ k L ( U ) + εµ ( U ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n I α ψ ε dµ − Z R n ψ ε dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ε (1 + C n,α,U ) µ ( R n ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n I α ψ ε dµ − Z R n ψ ε dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ε (1 + κ n,U ) µ ( R n )for all α ∈ (0 , ), so thatlim sup α → + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ϕ I α µ dx − Z R n ϕ dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (1 + κ n,U ) µ ( R n ) . The conclusion thus follows passing to the limit as ε → + . (cid:3) Theorem 4.9.
If either f ∈ BV ( R n ) or f ∈ bv ( R n ) ∩ L ∞ ( R n ) , then | D α f | ⇀ | Df | as α → − . (4.12) Moreover, if f ∈ BV ( R n ) , then also lim α → − | D α f | ( R n ) = | Df | ( R n ) . (4.13) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 43
Proof.
We prove (4.12) and (4.13) separately.
Proof of (4.12). By Theorem 4.7, we know that D α f ⇀ Df as α → − . By [22,Proposition 4.29], we thus have that | Df | ( A ) ≤ lim inf α → − | D α f | ( A ) (4.14)for any open set A ⊂ R n . Now let K ⊂ R n be a compact set. By the representationformula (3.18) in Corollary 3.6, we can estimate | D α f | ( K ) = k∇ α f k L ( K ; R n ) ≤ k I − α | Df |k L ( K ) = ( I − α | Df | L n )( K ) . Since | Df | ( R n ) < + ∞ , by Lemma 4.8 and [22, Proposition 4.26] we can conclude thatlim sup α → − | D α f | ( K ) ≤ lim sup α → − ( I − α | Df | L n )( K ) ≤ | Df | ( K ) , and so (4.12) follows, thanks again to [22, Proposition 4.26]. Proof of (4.13). Now assume f ∈ BV ( R n ). By (3.4) applied with A = R n and r = 1,we have | D α f | ( R n ) ≤ nω n µ n,α n + α − | Df | ( R n )1 − α + n + 2 α − α k f k L ( R n ) ! . By (4.2), we thus get that lim sup α → − | D α f | ( R n ) ≤ | Df | ( R n ) . (4.15)Thus (4.13) follows combining (4.14) for A = R n with (4.15). (cid:3) Note that Theorem 4.7 and Theorem 4.9 in particular apply to any f ∈ W , ( R n ).In the following result, by exploiting Proposition 3.3, we prove that a stronger propertyholds for any f ∈ W ,p ( R n ) with p ∈ (1 , + ∞ ). Theorem 4.10.
Let p ∈ (1 , + ∞ ) . If f ∈ W ,p ( R n ) , then lim α → − k∇ αw f − ∇ w f k L p ( R n ; R n ) = 0 . (4.16) Proof.
By Proposition 3.3 we know that f ∈ S α,p ( R n ) for any α ∈ (0 , Step 1 . We claim that lim α → − k∇ αw f k L p ( R n ; R n ) = k∇ w f k L p ( R n ; R n ) . (4.17)Indeed, on the one hand, by Proposition 4.4, we have Z R n ϕ · ∇ w f dx = − Z R n f div ϕ dx = − lim α → − Z R n f div α ϕ dx = lim α → − Z R n ϕ · ∇ αw f dx (4.18)for all ϕ ∈ C ∞ c ( R n ; R n ), so that Z R n ϕ · ∇ w f dx ≤ k ϕ k L pp − ( R n ; R n ) lim inf α → − k∇ αw f k L p ( R n ; R n ) for all ϕ ∈ C ∞ c ( R n ; R n ). We thus get that k∇ w f k L p ( R n ; R n ) ≤ lim inf α → − k∇ αw f k L p ( R n ; R n ) . (4.19) On the other hand, applying (3.10) with A = R n and r = 1, we have k∇ αw f k L p ( R n ; R n ) ≤ nω n µ n,α n + α − k∇ w f k L p ( R n ; R n ) − α + n + 2 α − α k f k L p ( R n ) ! . By (4.2), we conclude thatlim sup α → − k∇ αw f k L p ( R n ; R n ) ≤ k∇ w f k L p ( R n ; R n ) . (4.20)Thus, (4.17) follows combining (4.19) and (4.20). Step 2 . We now claim that ∇ αw f ⇀ ∇ w f in L p ( R n ; R n ) as α → − . (4.21)Indeed, let ϕ ∈ L pp − ( R n ; R n ). For each ε >
0, let ψ ε ∈ C ∞ c ( R n ; R n ) be such that k ψ ε − ϕ k L pp − ( R n ; R n ) < ε . By (4.18) and (4.17), we can estimatelim sup α → − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ϕ · ∇ αw f dx − Z R n ϕ · ∇ w f dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup α → − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ψ ε · ∇ αw f dx − Z R n ψ ε · ∇ w f dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Z R n | ϕ − ψ ε | |∇ αw f | dx + Z R n | ϕ − ψ ε | |∇ w f | dx ≤ ε lim α → − k∇ αw f k L p ( R n ; R n ) + k∇ w f k L p ( R n ; R n ) ! = 2 ε k∇ w f k L p ( R n ; R n ) so that (4.21) follows passing to the limit as ε → + .Since L p ( R n ; R n ) is uniformly convex (see [8, Section 4.3] for example), the limitin (4.16) follows from (4.17) and (4.21) by [8, Proposition 3.32], and the proof is com-plete. (cid:3) For the case p = + ∞ , we have the following result. Theorem 4.11. If f ∈ W , ∞ ( R n ) , then ∇ αw f ⇀ ∇ w f in L ∞ ( R n ; R n ) as α → − (4.22) and k∇ w f k L ∞ ( R n ; R n ) ≤ lim inf α → − k∇ αw f k L ∞ ( R n ; R n ) . (4.23) Proof.
We argue similarly as in the proof of Theorem 4.10, in two steps.
Step 1: proof of (4.22). By Proposition 2.8 and Proposition 4.4, we havelim α → − Z R n ϕ · ∇ α f dx = − lim α → − Z R n f div α ϕ dx = − Z R n f div ϕ dx = Z R n ϕ · ∇ w f dx (4.24)for all ϕ ∈ C ∞ c ( R n ; R n ), so that Z R n ϕ · ∇ w f dx ≤ k ϕ k L ( R n ; R n ) lim inf α → − k∇ α f k L ∞ ( R n ; R n ) for all ϕ ∈ C ∞ c ( R n ; R n ). We thus get (4.23). DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 45
Step 2: proof of (4.23). Let ϕ ∈ L ( R n ; R n ). For each ε >
0, let ψ ε ∈ C ∞ c ( R n ; R n ) besuch that k ψ ε − ϕ k L ( R n ; R n ) < ε . By (4.24) and (3.14), we can estimatelim sup α → − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ϕ · ∇ αw f dx − Z R n ϕ · ∇ w f dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup α → − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n ψ ε · ∇ αw f dx − Z R n ψ ε · ∇ w f dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Z R n | ϕ − ψ ε ||∇ αw f | dx + Z R n | ϕ − ψ ε ||∇ w f | dx ≤ ε lim sup α → − k∇ α f k L ∞ ( R n ; R n ) + k∇ f k L ∞ ( R n ; R n ) ! ≤ ε ( n + 1) k∇ w f k L ∞ ( R n ; R n ) so that (4.21) follows passing to the limit as ε → + . (cid:3) Remark 4.12.
We notice that Theorem 4.7 and Theorem 4.9, in the case f = χ E ∈ BV ( R n ) with E ⊂ R n bounded, and Theorem 4.10, were already announced in [34,Theorems 16 and 17].4.3. Γ -convergence of α -variation as α → − . In this section, we study the Γ-convergence of the fractional α -variation to the standard variation as α → − .We begin with the Γ - lim inf inequality. Theorem 4.13 (Γ - lim inf inequalities as α → − ) . Let Ω ⊂ R n be an open set.(i) If ( f α ) α ∈ (0 , ⊂ L ( R n ) satisfies sup α ∈ (0 , k f α k L ∞ ( R n ) < + ∞ and f α → f in L ( R n ) as α → − , then | Df | (Ω) ≤ lim inf α → − | D α f α | (Ω) . (4.25) (ii) If ( f α ) α ∈ (0 , ⊂ L ( R n ) satisfies f α → f in L ( R n ) as α → − , then (4.25) holds.Proof. We prove the two statements separately.
Proof of (i) . Let ϕ ∈ C ∞ c (Ω; R n ) be such that k ϕ k L ∞ (Ω; R n ) ≤
1. Since we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n f α div α ϕ dx − Z R n f div ϕ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n | f α − f | | div ϕ | dx + Z R n | f α | | div α ϕ − div ϕ | dx ≤ k div ϕ k L ∞ ( R n ; R n ) Z supp ϕ | f α − f | dx + (cid:16) sup α ∈ (0 , k f α k L ∞ ( R n ) (cid:17) k div α ϕ − div ϕ k L ( R n ) , by Proposition 4.4 we get that Z R n f div ϕ dx = lim α → − Z R n f α div α ϕ dx ≤ lim inf α → − | D α f | (Ω)and the conclusion follows. Proof of (ii) . Let ϕ ∈ C ∞ c (Ω; R n ) be such that k ϕ k L ∞ (Ω; R n ) ≤
1. Since we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n f α div α ϕ dx − Z R n f div ϕ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n | f α − f | | div ϕ | dx + Z R n | f α | | div α ϕ − div ϕ | dx ≤ k div ϕ k L ∞ ( R n ) k f α − f k L ( R n ) + k div α ϕ − div ϕ k L ∞ ( R n ) k f α k L ( R n ) , by Proposition 4.4 we get that Z R n f div ϕ dx = lim α → − Z R n f α div α ϕ dx ≤ lim inf α → − | D α f α | (Ω)and the conclusion follows. (cid:3) We now pass to the Γ - lim sup inequality.
Theorem 4.14 (Γ - lim sup inequalities as α → − ) . Let Ω ⊂ R n be an open set.(i) If f ∈ BV ( R n ) and either Ω is bounded or Ω = R n , then lim sup α → − | D α f | (Ω) ≤ | Df | (Ω) . (4.26) (ii) If f ∈ BV loc ( R n ) and Ω is bounded, then Γ( L ) - lim sup α → − | D α f | (Ω) ≤ | Df | (Ω) . In addition, if f = χ E , then the recovering sequences ( f α ) α ∈ (0 , in (i) and (ii) can betaken such that f α = χ E α for some measurable sets ( E α ) α ∈ (0 , .Proof. Assume f ∈ BV ( R n ). By Theorem 4.9, we know that | D α f | ⇀ | Df | as α → − .Thus, by [22, Proposition 4.26], we get thatlim sup α → − | D α f | (Ω) ≤ lim sup α → − | D α f | (Ω) ≤ | Df | (Ω) (4.27)for any bounded open set Ω ⊂ R n . If Ω = R n , then (4.26) follows immediately from (4.13).This concludes the proof of (i).Now assume that f ∈ BV loc ( R n ) and Ω is bounded. Let ( R k ) k ∈ N ⊂ (0 , + ∞ ) be asequence such that R k → + ∞ as k → + ∞ and set f k := f χ B Rk for all k ∈ N . ByTheorem A.1, we can choose the sequence ( R k ) k ∈ N such that, in addition, f k ∈ BV ( R n )with Df k = χ ⋆B Rk Df + f ⋆ Dχ B Rk for all k ∈ N . Consequently, f k → f in L ( R n ) as k → + ∞ and, moreover, since Ω is bounded, | Df k | (Ω) = | Df | (Ω) and | Df k | ( ∂ Ω) = | Df | ( ∂ Ω)for all k ∈ N sufficiently large. By (4.27), we have thatlim sup α → − | D α f k | (Ω) ≤ | Df k | (Ω) (4.28)for all k ∈ N sufficiently large. Hence, by [7, Proposition 1.28], by [12, Proposition 8.1(c)]and by (4.28), we get thatΓ( L ) - lim sup α → − | D α f | (Ω) ≤ lim inf k → + ∞ (cid:16) Γ( L ) - lim sup α → − | D α f k | (Ω) (cid:17) ≤ lim k → + ∞ | Df k | (Ω) = | Df | (Ω) . This concludes the proof of (ii).Finally, if f = χ E , then we can repeat the above argument verbatim in the metric spaces { χ F ∈ L ( R n ) : F ⊂ R n } for (i) and { χ F ∈ L ( R n ) : F ⊂ R n } for (ii) endowed with theirnatural distances. (cid:3) Remark 4.15.
Thanks to (4.26), a recovery sequence in Theorem 4.14(i) is the constantsequence (also in the special case f = χ E ).Combining Theorem 4.13(i) and Theorem 4.14(ii), we can prove that the fractionalCaccioppoli α -perimeter Γ-converges to De Giorgi’s perimeter as α → − in L ( R n ). Werefer to [2] for the same result on the classical fractional perimeter. Theorem 4.16 (Γ( L ) - lim of perimeters as α → − ) . Let Ω ⊂ R n be a bounded openset with Lipschitz boundary. For every measurable set E ⊂ R n , we have Γ( L ) - lim α → − | D α χ E | (Ω) = P ( E ; Ω) . DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 47
Proof.
By Theorem 4.13(i), we already know thatΓ( L ) - lim inf α → − | D α χ E | (Ω) ≥ P ( E ; Ω) , so we just need to prove the Γ( L ) - lim sup inequality. Without loss of generality, wecan assume P ( E ; Ω) < + ∞ . Now let ( E k ) k ∈ N be given by Theorem A.4. Since χ E k ∈ BV loc ( R n ) and P ( E k ; ∂ Ω) = 0 for all k ∈ N , by Theorem 4.14(ii) we know thatΓ( L ) - lim sup α → − | D α χ E k | (Ω) ≤ P ( E k ; Ω)for all k ∈ N . Since χ E k → χ E in L ( R n ) and P ( E k ; Ω) → P ( E ; Ω) as k → + ∞ ,by [7, Proposition 1.28] we get thatΓ( L ) - lim sup α → − | D α χ E | (Ω) ≤ lim inf k → + ∞ (cid:16) Γ( L ) - lim sup α → − | D α χ E k | (Ω) (cid:17) ≤ lim k → + ∞ P ( E k ; Ω) = P ( E ; Ω)and the proof is complete. (cid:3) Finally, combining Theorem 4.13(ii) and Theorem 4.14, we can prove that the fractional α -variation Γ-converges to De Giorgi’s variation as α → − in L ( R n ). Theorem 4.17 (Γ( L ) - lim of variations as α → − ) . Let Ω ⊂ R n be an open set suchthat either Ω is bounded with Lipschitz boundary or Ω = R n . For every f ∈ BV ( R n ) , wehave Γ( L ) - lim α → − | D α f | (Ω) = | Df | (Ω) . Proof.
The case Ω = R n follows immediately by [12, Proposition 8.1(c)] combining The-orem 4.13(ii) with Theorem 4.14(i). We can thus assume that Ω is a bounded openset with Lipschitz boundary and argue similarly as in the proof of Theorem 4.16. ByTheorem 4.13(ii), we already know thatΓ( L ) - lim inf α → − | D α f | (Ω) ≥ | Df | (Ω) , so we just need to prove the Γ( L ) - lim sup inequality. Without loss of generality, we canassume | Df | (Ω) < + ∞ . Now let ( f k ) k ∈ N ⊂ BV ( R n ) be given by Theorem A.6. Since | Df k | ( ∂ Ω) = 0 for all k ∈ N , by Theorem 4.14 we know thatΓ( L ) - lim sup α → − | D α f k | (Ω) ≤ | Df k | (Ω) = | Df k | (Ω)for all k ∈ N . Since f k → f in L ( R n ) and | D α f k | (Ω) → | D α f | (Ω) as k → + ∞ ,by [7, Proposition 1.28] we get thatΓ( L ) - lim sup α → − | D α f | (Ω) ≤ lim inf k → + ∞ (cid:16) Γ( L ) - lim sup α → − | D α f k | (Ω) (cid:17) ≤ lim k → + ∞ | Df k | (Ω) = | Df | (Ω)and the proof is complete. (cid:3) Remark 4.18.
Thanks to Theorem 4.17, we can slightly improve Theorem 4.16. Indeed,if χ E ∈ BV ( R n ), then we also haveΓ( L ) - lim α → − | D α χ E | (Ω) = | Dχ E | (Ω) for any open set Ω ⊂ R n such that either Ω is bounded with Lipschitz boundary or Ω = R n .5. Asymptotic behaviour of fractional β -variation as β → α − Convergence of ∇ β and div β as β → α . We begin with the following simple resultabout the L -convergence of the operators ∇ β and div β as β → α with α ∈ (0 , Lemma 5.1.
Let α ∈ (0 , . If f ∈ W α, ( R n ) and ϕ ∈ W α, ( R n ; R n ) , then lim β → α − k∇ β f − ∇ α f k L ( R n ; R n ) = 0 , lim β → α − k div β ϕ − div α ϕ k L ( R n ) = 0 . (5.1) Proof.
Given β ∈ (0 , α ), we can estimate Z R n |∇ β f ( x ) − ∇ α f ( x ) | dx ≤ | µ n,β − µ n,α | [ f ] W α, ( R n ) + µ n,β Z R n Z R n | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy dx. Since the Γ function is continuous (see [4]), we clearly havelim β → α − | µ n,β − µ n,α | [ f ] W α, ( R n ) = 0 . Now write Z R n Z R n | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy dx = Z R n Z R n | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ (0 , ( | y − x | ) dy dx + Z R n Z R n | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ [1 , + ∞ ) ( | y − x | ) dy dx. On the one hand, since f ∈ W α, ( R n ), we have | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ (0 , ( | y − x | )= | f ( y ) − f ( x ) || y − x | n | y − x | α − | y − x | β ! χ (0 , ( | y − x | ) ≤ | f ( y ) − f ( x ) || y − x | n + α χ (0 , ( | y − x | ) ∈ L x,y ( R n )and thus, by Lebesgue’s Dominated Convergence Theorem, we get thatlim β → α − Z R n Z R n | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ (0 , ( | y − x | ) dy dx = 0 . On the other hand, since one has[ f ] W β, ( R n ) = Z R n Z {| h | < } | f ( x + h ) − f ( x ) || h | n + β dh dx + Z R n Z {| h |≥ } | f ( x + h ) − f ( x ) || h | n + β dh dx ≤ [ f ] W α, ( R n ) + Z {| h |≥ } | h | n + β Z R n | f ( x + h ) | + | f ( x ) | dx dh DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 49 = [ f ] W α, ( R n ) + 2 nω n β k f k L ( R n ) for all β ∈ (0 , α ), we can estimate | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ [1 , + ∞ ) ( | y − x | )= | f ( y ) − f ( x ) || y − x | n | y − x | β − | y − x | α ! χ [1 , + ∞ ) ( | y − x | ) ≤ | f ( y ) − f ( x ) || y − x | n + β χ [1 , + ∞ ) ( | y − x | ) ≤ | f ( y ) − f ( x ) || y − x | n + α χ [1 , + ∞ ) ( | y − x | ) ∈ L x,y ( R n )for all β ∈ (cid:16) α , α (cid:17) and thus, by Lebesgue’s Dominated Convergence Theorem, we get thatlim β → α − Z R n Z R n | f ( y ) − f ( x ) || y − x | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | y − x | β − | y − x | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ [1 , + ∞ ) ( | y − x | ) dy dx = 0and the first limit in (5.1) follows. The second limit in (5.1) follows similarly and we leavethe proof to the reader. (cid:3) Remark 5.2.
Let α ∈ (0 , f ∈ W α + ε, ( R n ) and ϕ ∈ W α + ε, ( R n ) for some ε ∈ (0 , − α ), then, arguing as in the proof of Lemma 5.1, one can also prove thatlim β → α + k∇ β f − ∇ α f k L ( R n ; R n ) = 0 , lim β → α + k div β ϕ − div α ϕ k L ( R n ) = 0 . We leave the details of proof of this result to the interested reader.If one deals with more regular functions, then Lemma 5.1 can be improved as follows.
Lemma 5.3.
Let α ∈ (0 , and p ∈ [1 , + ∞ ] . If f ∈ Lip c ( R n ) and ϕ ∈ Lip c ( R n ; R n ) , then lim β → α − k∇ β f − ∇ α f k L p ( R n ; R n ) = 0 , lim β → α − k div β ϕ − div α ϕ k L p ( R n ) = 0 . (5.2) Proof.
Since clearly f ∈ W α, ( R n ) for any α ∈ (0 , p = 1 follows from Lemma 5.1. Hence, we just need to prove the validity of the samelimit for the case p = + ∞ , since then the conclusion simply follows by an interpolationargument.Let β ∈ (0 , α ) and x ∈ R n . We have |∇ α f ( x ) − ∇ β f ( x ) | ≤ | µ n,β − µ n,α | Z R n | f ( x ) − f ( y ) || x − y | n + α dy + µ n,β Z R n | f ( x ) − f ( y ) || x − y | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | x − y | β − | x − y | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy = | µ n,β − µ n,α | Z R n | f ( x + z ) − f ( x ) || z | n + α dz + µ n,β Z R n | f ( x + z ) − f ( x ) || z | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | z | β − | z | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz. Since Z R n | f ( x + z ) − f ( x ) || z | n + α dz ≤ Z {| z |≤ } Lip( f ) | z | n + α − dz + Z {| z | > } k f k L ∞ ( R n ) | z | n + α dz ≤ nω n Lip( f )1 − α + 2 k f k L ∞ ( R n ) α ! and Z R n | f ( x + z ) − f ( z ) || z | n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | z | β − | z | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz ≤ Z {| z |≤ } Lip( f ) | z | n − | z | α − | z | β ! dz + Z {| z | > } k f k L ∞ ( R n ) | z | n | z | β − | z | α ! dz ≤ ( α − β ) nω n Lip( f )(1 − α )(1 − β ) + 2 k f k L ∞ ( R n ) αβ ! , for all β ∈ (cid:16) α , α (cid:17) we obtain k∇ α f − ∇ β f k L ∞ ( R n ; R n ) ≤ c n,α max n Lip( f ) , k f k L ∞ ( R n ) o (cid:16) | µ n,β − µ n,α | + ( α − β ) (cid:17) , for some constant c n,α > n and α . Thus the conclusion follows since µ n,β → µ n,α as β → α − . The second limit in (5.2) follows similarly and we leave the proofto the reader. (cid:3) Weak convergence of β -variation as β → α − . In Theorem 5.4 below, we provethe weak convergence of the β -variation as β → α − , extending the convergences obtainedin Theorem 4.7 and Theorem 4.9. Theorem 5.4.
Let α ∈ (0 , . If f ∈ BV α ( R n ) , then D β f ⇀ D α f and | D β f | ⇀ | D α f | as β → α − . Moreover, we have lim β → α − | D β f | ( R n ) = | D α f | ( R n ) . (5.3) Proof.
We divide the proof in three steps.
Step 1: we prove that D β f ⇀ D α f as β → α − . We argue similarly as in Step 1 of theproof of Theorem 4.7. By Proposition 3.12, we have Z R n ϕ · ∇ β f dx = − Z R n f div β ϕ dx for all β ∈ (0 , α ) and ϕ ∈ Lip c ( R n ; R n ). Thus, thanks to (5.2) in the case p = ∞ , we getlim β → α − Z R n ϕ · ∇ β f dx = − lim β → α − Z R n f div β ϕ dx = − Z R n f div α ϕ dx = Z R n ϕ · dD α f. Now fix ϕ ∈ C c ( R n ; R n ). Let U ⊂ R n be a fixed bounded open set such that supp ϕ ⊂ U .For each ε > ψ ε ∈ Lip c ( R n ; R n ) such that k ϕ − ψ ε k L ∞ ( R n ; R n ) < ε DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 51 and supp ψ ε ⊂ U . Then, by (3.25), we can estimate (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ β f dx − Z R n ϕ · dD α f (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ − ψ ε k L ∞ ( R n ; R n ) (cid:18)Z U |∇ β f | dx + | D α f | ( R n ) (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12)Z R n ψ ε · ∇ β f dx − Z R n ψ ε · dD α f (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (1 + C n, (1 − α + β ) ,U ) | D α f | ( R n )+ (cid:12)(cid:12)(cid:12)(cid:12)Z R n ψ ε · ∇ α f dx − Z R n ψ ε · dDf (cid:12)(cid:12)(cid:12)(cid:12) for all β ∈ (0 , α ). Thus, by the uniform estimate (4.3) in Lemma 4.2, we getlim β → α − (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ · ∇ α f dx − Z R n ϕ · dDf (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (1 + κ n,U ) | D α f | ( R n ) (5.4)and the conclusion follows passing to the limit as ε → + . Step 2: we prove that | D β f | ⇀ | D α f | as β → α − . We argue similarly as in the firstpart of the proof of Theorem 4.9. Since D β f ⇀ D α f as β → α − as proved in Step 1above, by [22, Proposition 4.29], we have that | D α f | ( A ) ≤ lim inf β → α − | D β f | ( A ) (5.5)for any open set A ⊂ R n . Now let K ⊂ R n be a compact set. By the representationformula (3.24) in Proposition 3.12, we can estimate | D β f | ( K ) = k∇ β f k L ( K ; R n ) ≤ k I α − β | D α f |k L ( K ) = ( I α − β | D α f | L n )( K ) . Since | D α f | ( R n ) < + ∞ , by Lemma 4.8 and [22, Proposition 4.26] we conclude thatlim sup β → α − | D β f | ( K ) ≤ lim sup β → α − ( I α − β | D α f | L n )( K ) ≤ | D α f | ( K ) . (5.6)The conclusion thus follows thanks to [22, Proposition 4.26]. Step 3: we prove (5.3). We argue similarly as in the proof of (4.12). By (3.26) appliedwith A = R n and r = 1, we have | D β f | ( R n ) ≤ µ n, β − α n + β − α nω n α − β | D α f | ( R n ) + ω n,α ( n + 2 β − α ) β k f k L ( R n ) ! . By (4.2), we get that lim sup β → α − | D β f | ( R n ) ≤ | D α f | ( R n ) . (5.7)Thus, (5.3) follows combining (5.5) for A = R n with (5.7). (cid:3) -convergence of β -variation as β → α − . In this section, we study the Γ-con-vergence of the fractional β -variation as β → α − , partially extending the results obtainedin Section 4.3.We begin with the Γ - lim inf inequality. Theorem 5.5 (Γ - lim inf inequality for β → α − ) . Let α ∈ (0 , and let Ω ⊂ R n be anopen set. If ( f β ) β ∈ (0 ,α ) ⊂ L ( R n ) satisfies f β → f in L ( R n ) as β → α − , then | D α f | (Ω) ≤ lim inf β → α − | D β f β | (Ω) . (5.8) Proof.
We argue similarly as in the proof of Theorem 4.13(ii). Let ϕ ∈ C ∞ c (Ω; R n ) besuch that k ϕ k L ∞ (Ω; R n ) ≤
1. Let U ⊂ R n be a bounded open set such that supp ϕ ⊂ U .By (2.12), we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R n f β div β ϕ dx − Z R n f div α ϕ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n | f β − f | | div β ϕ | dx + Z R n | f | | div β ϕ − div α ϕ | dx ≤ C n,β,U k div ϕ k L ∞ ( R n ; R n ) k f β − f k L ( R n ) + Z R n | f | | div β ϕ − div α ϕ | dx for all β ∈ (0 , α ). Since div β ϕ → div α ϕ in L ∞ ( R n ) as β → α − by (5.2), we easily obtainlim β → α − Z R n | f | | div β ϕ − div α ϕ | dx = 0 . Hence, we get Z R n f div α ϕ dx = lim β → α − Z R n f β div β ϕ dx ≤ lim inf β → α − | D β f β | (Ω)and the conclusion follows. (cid:3) We now pass to the Γ - lim sup inequality.
Theorem 5.6 (Γ - lim sup inequality for β → α − ) . Let α ∈ (0 , and let Ω ⊂ R n be anopen set. If f ∈ BV α ( R n ) and either Ω is bounded or Ω = R n , then lim sup β → α − | D β f | (Ω) ≤ | D α f | (Ω) . (5.9) Proof.
We argue similarly as in the proof of Theorem 4.14. By Theorem 5.4, we knowthat | D β f | ⇀ | D α f | as β → α − . Thus, by [22, Proposition 4.26] and (5.3), we get thatlim sup β → α − | D β f | (Ω) ≤ lim sup β → α − | D β f | (Ω) ≤ | D α f | (Ω) (5.10)for any open set Ω ⊂ R n such that either Ω is bounded or Ω = R n . (cid:3) Corollary 5.7 (Γ( L ) - lim of variations in R n as β → α − ) . Let α ∈ (0 , . For every f ∈ BV α ( R n ) , we have Γ( L ) - lim β → α − | D β f | ( R n ) = | D α f | ( R n ) . In particular, the constant sequence is a recovery sequence.Proof.
The result follows easily by combining (5.8) and (5.9) in the case Ω = R n . (cid:3) Remark 5.8.
We recall that, by [10, Theorem 3.25], f ∈ BV α ( R n ) satisfies | D α f | ≪ L n if and only if f ∈ S α, ( R n ). Therefore, if f ∈ S α, ( R n ), then | D α f | ( ∂ Ω) = 0 for anybounded open set Ω ⊂ R n such that L n ( ∂ Ω) = 0 (for instance, Ω with Lipschitzboundary). Thus, we can actually obtain the Γ-convergence of the fractional β -variationas β → α − on bounded open sets with Lipschitz boundary for any f ∈ S α, ( R n ) too. In-deed, it is enough to combine (5.8) and (5.9) and then exploit the fact that | D α f | ( ∂ Ω) = 0to get Γ( L ) - lim β → α − | D β f | (Ω) = | D α f | (Ω)for any f ∈ S α, ( R n ). DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 53
Appendix A. Truncation and approximation of BV functions For the reader’s convenience, in this appendix we state and prove two known results on BV functions and sets with locally finite perimeter.A.1. Truncation of BV functions. Following [3, Section 3.6] and [15, Section 5.9],given f ∈ L ( R n ), we define its precise representative f ⋆ : R n → [0 , + ∞ ] as f ⋆ ( x ) := lim r → + ω n r n Z B r ( x ) f ( y ) dy, x ∈ R n , (A.1)if the limit exists, otherwise we let f ⋆ ( x ) = 0 by convention. Theorem A.1 (Truncation of BV functions) . If f ∈ BV loc ( R n ) , then f χ B r ∈ BV ( R n ) , with D ( f χ B r ) = χ ⋆B r Df + f ⋆ Dχ B r , (A.2) for L -a.e. r > . If, in addition, f ∈ L ∞ ( R n ) , then (A.2) holds for all r > .Proof. Fix ϕ ∈ C ∞ c ( R n ; R n ) and let U ⊂ R n be a bounded open set such that supp( ϕ ) ⊂ U .Let ( ̺ ε ) ε> ⊂ C ∞ c ( R n ) be a family of standard mollifiers as in [10, Section 3.3] and set f ε := f ∗ ̺ ε for all ε >
0. Note that supp (cid:16) ̺ ε ∗ ( χ B r ϕ ) (cid:17) ⊂ U and supp (cid:16) ̺ ε ∗ ( χ B r div ϕ ) (cid:17) ⊂ U for all ε > r >
0. Given r >
0, by Leibniz’s rule and Fubini’sTheorem, we have Z R n f ε χ B r div ϕ dx = Z R n χ B r div( f ε ϕ ) dx − Z R n χ B r ϕ · ∇ f ε dx = − Z R n f ε ϕ · dDχ B r − Z R n ̺ ε ∗ ( χ B r ϕ ) · dDf. (A.3)Since f ε → f a.e. in R n as ε → + and | f | ̺ ε ∗ ( χ B r | div ϕ | ) ≤ | f | χ U k div ϕ k L ∞ ( R n ) ∈ L ( R n )for all ε >
0, by Lebesgue’s Dominated Convergence Theorem we havelim ε → + Z R n f ε χ B r div ϕ dx = Z R n f χ B r div ϕ dx for all r >
0. Thus, since ̺ ε ∗ ( χ B r ϕ ) → χ ⋆B r ϕ pointwise in R n as ε → + and | ̺ ε ∗ ( χ B r ϕ ) | ≤ k ϕ k L ∞ ( R n ; R n ) χ U ∈ L ( R n , | Df | )for all ε > ε → + Z R n ̺ ε ∗ ( χ B r ϕ ) · dDf = Z R n χ ⋆B r ϕ · dDf for all r >
0. Now, by [3, Theorem 3.78 and Corollary 3.80], we know that f ε → f ⋆ H n − -a.e. in R n as ε → + . As a consequence, given any r >
0, we get that f ε → f ⋆ | Dχ B r | -a.e. in R n as ε → + . Thus, if f ∈ L ∞ ( R n ), then | f ε ϕ | ≤ k f k L ∞ ( R n ) | ϕ | ∈ L ( R n , | Dχ B r | )for all ε > ε → + Z R n f ε ϕ · dDχ B r = Z R n f ⋆ ϕ · dDχ B r for all r >
0. Therefore, if f ∈ L ∞ ( R n ), then we can pass to the limit as ε → + in (A.3)and get Z R n f χ B r div ϕ dx = − Z R n f ⋆ ϕ · dDχ B r − Z R n χ ⋆B r ϕ · dDf for all ϕ ∈ C ∞ c ( R n ; R n ) and for all r >
0. Since k f ⋆ k L ∞ ( R n ) ≤ k f k L ∞ ( R n ) , this proves (A.2)for all r >
0. If f is not necessarily bounded, then we argue as follows. Without loss ofgenerality, assume that k ϕ k L ∞ ( R n ; R n ) ≤
1. We can thus estimate (cid:12)(cid:12)(cid:12)(cid:12)Z R n f ε ϕ · dDχ B r − Z R n f ⋆ ϕ · dDχ B r (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∂B r | f ε − f ⋆ | d H n − . (A.4)Given any R >
0, by Fatou’s Lemma we thus get that Z R lim inf ε → + (cid:12)(cid:12)(cid:12)(cid:12)Z R n f ε ϕ · dDχ B r − Z R n f ⋆ ϕ · dDχ B r (cid:12)(cid:12)(cid:12)(cid:12) dr ≤ Z R lim inf ε → + Z ∂B r | f ε − f ⋆ | d H n − dr ≤ lim inf ε → + Z R Z ∂B r | f ε − f ⋆ | d H n − dr = lim ε → + Z B R | f ε − f ⋆ | dx = 0 . Hence, the set Z := (cid:26) r > ε → + Z ∂B r | f ε − f ⋆ | d H n − = 0 (cid:27) (A.5)satisfies L ((0 , + ∞ ) \ Z ) = 0 and depends neither on the choice of ϕ nor on the choiceof the L n -representative of f . Now fix r ∈ Z and let ( ε k ) k ∈ N be any sequence realisingthe lim inf in (A.5). By (A.4), we thus getlim k → + ∞ Z R n f ε k ϕ · dDχ B r = Z R n f ⋆ ϕ · dDχ B r uniformly for all ϕ satisfying k ϕ k L ∞ ( R n ; R n ) ≤
1. Passing to the limit along the sequence( ε k ) k ∈ N as k → + ∞ in (A.3), we get that Z R n f χ B r div ϕ dx = − Z R n f ⋆ ϕ · dDχ B r − Z R n χ ⋆B r ϕ · dDf for all ϕ ∈ C ∞ c ( R n ; R n ) with k ϕ k L ∞ ( R n ; R n ) ≤
1. Finally, since Z R Z ∂B r | f ⋆ | d H n − dr = Z B R | f ⋆ | dx < + ∞ , the set W := (cid:26) r > Z ∂B r | f ⋆ | d H n − dr < + ∞ (cid:27) satisfies L ((0 , + ∞ ) \ W ) = 0 and does not depend on the choice of the L n -representativeof f . Thus (A.2) follows for all r ∈ W ∩ Z and the proof is concluded. (cid:3) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 55
A.2.
Approximation by sets with polyhedral boundary.
In this section we stateand prove standard approximation results for sets with finite perimeter or, more generally, BV loc ( R n ) functions, in a sufficiently regular bounded open set.We need the following two preliminary lemmas. Lemma A.2.
Let
V, W ⊂ S n − , with V finite and W at most countable. For any ε > ,there exists R ∈
SO( n ) with |R − I| < ε , where I is the identity matrix, such that R ( V ) ∩ W = ∅ .Proof. Let N ∈ N be such that V = { v i ∈ S n − : i = 1 , . . . , N } . We divide the proof intwo steps. Step 1 . Assume that W is finite and set A i := {R ∈ SO( n ) : R ( v i ) / ∈ W } for all i =1 , . . . , N . We now claim that A i is an open and dense subset of SO( n ) for all i = 1 , . . . , N .Indeed, given any i = 1 , . . . , N , since W is finite, the set A ci = SO( n ) \ A i is closedin SO( n ). Moreover, we claim that int( A ci ) = ∅ . Indeed, by contradiction, let us assumethat int( A ci ) = ∅ . Then there exist ε > R ∈ A ci such that any S ∈
SO( n ) with |S − R| < ε satisfies S ∈ A ci . In particular, for these R ∈ A ci and ε >
0, we have R + ε k I|I| ∈ A ci for any k ≥
1, which implies R ( v i ) + ε k |I| v i ∈ W for any k ≥
1, in contrastwith the fact that W is finite. Thus, A i is an open and dense subset of SO( n ) for all i = 1 , . . . , N , and so also the set A W := N \ i =1 A i = {R ∈ SO( n ) : R ( v i ) / ∈ W ∀ i = 1 , . . . , N } is an open and dense subset of SO( n ). The result is thus proved for any finite set W . Step 2 . Now assume that W is countable, W = { w k ∈ S n − : k ∈ N } . For all M ∈ N ,set W M := { w k ∈ W : k ≤ M } . By Step 1, we know that A W M is an open and densesubset of SO( n ) for all M ∈ N . Since SO( n ) ⊂ R n is compact, by Baire’s Theorem A := T M ∈ N A W M is a dense subset of SO( n ). This concludes the proof. (cid:3) Since det : GL( n ) → R is a continuous map, there exists a dimensional constant δ n ∈ (0 ,
1) such that det
R ≥ for all R ∈
GL( n ) with |R − I| < δ n . Lemma A.3.
Let ε ∈ (0 , δ n ) and let E ⊂ R n be a bounded set with P ( E ) < + ∞ . If R ∈
SO( n ) satisfies |R − I| < ε , then |R ( E ) △ E | ≤ εr E P ( E ) , where r E := sup { r > | E \ B r | > } .Proof. We divide the proof in two steps.
Step 1 . Let r > f ∈ C ∞ c ( R n ). Setting R t := (1 − t ) I + t R for all t ∈ [0 , Z B r | f ( R ( x )) − f ( x ) | dx = Z B r (cid:12)(cid:12)(cid:12)(cid:12)Z h∇ f ( R t ( x )) , R ( x ) − x i dt (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ |R − I| r Z Z B r |∇ f ( R t ( x )) | dx dt. Since |R t − I| = t |R − I| < tε < δ n for all t ∈ [0 , R t is invertible with det( R − t ) ≤ t ∈ [0 , Z B r |∇ f ( R t ( x )) | dx = Z R t ( B r ) |∇ f ( y ) | | det( R − t ) | dy ≤ Z R n |∇ f ( y ) | dy, so that Z B r | f ( R ( x )) − f ( x ) | dx ≤ εr k∇ f k L ( R n ; R n ) . (A.6) Step 2 . Since χ E ∈ BV ( R n ), combining [15, Theorem 5.3] with a standard cut-offapproximation argument, we find ( f k ) k ∈ N ⊂ C ∞ c ( R n ) such that f k → χ E pointwise a.e.in R n and |∇ f k | ( R n ) → P ( E ) as k → + ∞ . Given any r >
0, by (A.6) in Step 1 we have Z B r | f k ( R ( x )) − f k ( x ) | dx ≤ εr k∇ f k k L ( R n ; R n ) for all k ∈ N . Passing to the limit as k → + ∞ , by Fatou’s Lemma we get that | ( R ( E ) △ E ) ∩ B r | ≤ εr P ( E ) . Since E ⊂ B r E up to L n -negligible sets, also R ( E ) ⊂ B r E up to L n -negligible sets. Thuswe can choose r = r E and the proof is complete. (cid:3) We are now ready to prove the main approximation result, see also [2, Proposition 15].
Theorem A.4.
Let Ω ⊂ R n be a bounded open set with Lipschitz boundary and let E ⊂ R n be a measurable set such that P ( E ; Ω) < + ∞ . There exists a sequence ( E k ) k ∈ N of boundedopen sets with polyhedral boundary such that P ( E k ; ∂ Ω) = 0 (A.7) for all k ∈ N and χ E k → χ E in L ( R n ) and P ( E k ; Ω) → P ( E ; Ω) (A.8) as k → + ∞ .Proof. We divide the proof in four steps.
Step 1: cut-off . Since Ω is bounded, we find R > ⊂ B R . Let us define R k = R + k and C k := (cid:26) x ∈ Ω c : dist( x, ∂ Ω) ≤ k (cid:27) for all k ∈ N . We set E k := E ∩ B R k ∩ C ck for all k ∈ N . Note that E k is a boundedmeasurable set such that χ E k → χ E in L ( R n ) as k → + ∞ and P ( E k ; Ω) = P ( E ; Ω) for all k ∈ N . Step 2: extension . Let us define A k := (cid:26) x ∈ R n : dist( x, Ω) < k (cid:27) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 57 for all k ∈ N . Since χ E k ∩ Ω ∈ BV (Ω) for all k ∈ N , by [3, Definition 3.20 and Proposi-tion 3.21] there exists a sequence ( v k ) k ∈ N ⊂ BV ( R n ) such that v k = 0 a.e. in A ck , v k = χ E k in Ω , | Dv k | ( ∂ Ω) = 0for all k ∈ N . Let us define F tk := { v k > t } for all t ∈ (0 , k ∈ N , by the coareaformula [3, Theorem 3.40], for a.e. t ∈ (0 ,
1) the set F tk has finite perimeter in R n andsatisfies F tk ⊂ A k , F tk ∩ Ω = E k ∩ Ω , P ( F tk ; ∂ Ω) = 0for all k ∈ N . We choose any such t k ∈ (0 ,
1) for each k ∈ N and define E k := E k ∪ F t k k for all k ∈ N . Note that E k is a bounded set with finite perimeter in R n such that χ E k → χ E in L ( R n ) as k → + ∞ and P ( E k ; Ω) = P ( E ; Ω) and P ( E k ; ∂ Ω) = 0 for all k ∈ N . Step 3: approximation . Let us define D k := (cid:26) x ∈ Ω c : dist( x, ∂ Ω) ∈ (cid:20) k , k (cid:21)(cid:27) for all k ∈ N . First arguing as in the first part of the proof of [22, Theorem 13.8] taking [22,Remark 13.13] into account, and then performing a standard diagonal argument, we finda sequence of bounded open sets ( E k ) k ∈ N with polyhedral boundary such that E k ⊂ D ck for all k ∈ N and χ E k → χ E in L ( R n ) , P ( E k ; Ω) → P ( E ; Ω) and P ( E k ; ∂ Ω) → k → + ∞ . If there exists a subsequence ( E k j ) j ∈ N such that P ( E k j ; ∂ Ω) = 0 for all j ∈ N , then we can set E j := E k j for all j ∈ N and the proof is concluded. If this is notthe case, then we need to proceed with the next last step. Step 4: rotation . We now argue as in the last part of the proof of [2, Proposition 15].Fix k ∈ N and assume P ( E k ; ∂ Ω) >
0. Since E k has polyhedral boundary, we have H n − ( ∂E k ∩ ∂ Ω) > ν ∈ S n − and U ⊂ F Ω such that H n − ( U ) > ν Ω ( x ) = ν for all x ∈ U and U ⊂ ∂H for some half-space H satisfying ν H = ν . Since P (Ω) = H n − ( ∂ Ω) < + ∞ , the set W : = n ν ∈ S n − : H n − ( { x ∈ ∂ Ω : ν Ω ( x ) = ν } ) > o = [ h ∈ N n ν ∈ S n − : P (Ω) h ≥ H n − ( { x ∈ ∂ Ω : ν Ω ( x ) = ν } ) > P (Ω) h +1 ) o is at most countable. Since E k has polyhedral boundary, the set V k := n ν ∈ S n − : H n − (cid:16)n x ∈ ∂E k : ν E k ( x ) = ν o(cid:17) > o is finite. By Lemma A.2, given ε k >
0, there exists R k ∈ SO( n ) with |R k − I| < ε k suchthat R k ( V k ) ∩ W = ∅ . Hence the set E k := R k ( E k ) must satisfy P ( E k ; ∂ Ω) = 0. ByLemma A.3, we can choose ε k > | E k △ E k | < k .Now choose η k ∈ (cid:16) , k (cid:17) such that P ( E k ; Q k ) ≤ P ( E k ; ∂ Ω), where Q k := { x ∈ R n : dist( x, ∂ Ω) < η k } . Since Ω is bounded, possibly choosing ε k > △ R − (Ω) ⊂ Q k . Hence we can estimate | P ( E k ; Ω) − P ( E k ; Ω) | = | H n − ( ∂E k ∩ R − (Ω)) − H n − ( ∂E k ∩ Ω) |≤ H n − (cid:16) ∂E k ∩ (Ω △ R − (Ω)) (cid:17) ≤ H n − ( ∂E k ∩ Q k ) . We can thus set E k := E k for all k ∈ N and the proof is complete. (cid:3) Remark A.5 (A minor gap in the proof of [2, Proposition 15]) . We warn the readerthat the cut-off and the extension steps presented above were not mentioned in the proofof [2, Proposition 15], although they are unavoidable for the correct implementation ofthe rotation argument in the last step. Indeed, in general, one cannot expect the existenceof a rotation
R ∈
SO( n ) arbitrarily close to the identity map such that P ( R ( E ); ∂ Ω) = 0and, at the same time, the difference between P ( R ( E ); Ω) and P ( E ; Ω) is small. Forexample, one can consider Ω = n ( x , x ) ∈ A : x + x < o and E = n ( x , x ) ∈ A : 1 < x + x < o ∪ n ( x , x ) ∈ A c : 9 < x + x < o where A = { ( x , x ) ∈ R : x > , x > } . In this case, for any rotation R ∈
SO(2)arbitrarily close to the identity map, we have P ( R ( E ); Ω) > P ( E ; Ω).We conclude this section with the following result, establishing an approximation of BV loc functions similar to that given in Theorem A.4. Theorem A.6.
Let Ω ⊂ R n be a bounded open set with Lipschitz boundary and let f ∈ BV loc ( R n ) . There exists ( f k ) k ∈ N ⊂ BV ( R n ) such that | Df k | ( ∂ Ω) = 0 for all k ∈ N and f k → f in L ( R n ) and | Df k | (Ω) → | Df | (Ω) as k → + ∞ . If, in addition, f ∈ L ( R n ) , then f k → f in L ( R n ) as k → + ∞ .Proof. We argue similarly as in the proof of Theorem A.4, in two steps.
Step 1: cut-off at infinity . Since Ω is bounded, we find R > ⊂ B R .Given ( R k ) k ⊂ ( R , + ∞ ), we set g k := f χ B Rk for all k ∈ N . By Theorem A.1, we have g k ∈ BV ( R n ) for a suitable choice of the sequence ( R k ) k ∈ N , with | Dg k | (Ω) = | Df | (Ω) forall k ∈ N and g k → f in L ( R n ) as k → + ∞ . If, in addition, f ∈ L ( R n ), then g k → f in L ( R n ) as k → + ∞ . Step 2: extension and cut-off near
Ω. Let us define A k := (cid:26) x ∈ R n : dist( x, Ω) < k (cid:27) DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 59 for all k ∈ N . Since g k χ Ω ∈ BV (Ω) with | Dg k | (Ω) = | Df | (Ω) for all k ∈ N , by [3,Definition 3.20 and Proposition 3.21] there exists a sequence ( h k ) k ∈ N ⊂ BV ( R n ) suchthat supp h k ⊂ A k , h k = g k in Ω , | Dh k | ( ∂ Ω) = 0for all k ∈ N and lim k → + ∞ Z A k \ Ω | h k | dx = 0(the latter property easily follows from the construction performed in the proof of [3,Proposition 3.21]). Now let ( v k ) k ∈ N ⊂ C ∞ c ( R n ) be such that supp v k ⊂ A ck and 0 ≤ v k ≤ k ∈ N and v k → χ Ω c pointwise in R n as k → + ∞ . We can thus set f k := h k + v k g k for all k ∈ N . By [3, Propositon 3.2(b)], we have v k g k ∈ BV ( R n ) for all k ∈ N , so that f k ∈ BV ( R n ) for all k ∈ N . Since we can estimate | f k − f | ≤ | h k − f χ Ω | + | v k − χ Ω c | | g k | + | g k − f | χ Ω c = | h k | χ A k \ Ω + | v k − χ Ω c | | g k | + | g k − f | χ Ω c for all k ∈ N , we have f k → f in L ( R n ) as k → + ∞ , with f k → f in L ( R n ) as k → + ∞ if f ∈ L ( R n ). By construction, we also have | Df k | (Ω) = | Dh k | (Ω) and | Df k | ( ∂ Ω) = | Dh k | ( ∂ Ω)for all k ∈ N . The proof is complete. (cid:3) References [1] D. R. Adams and L. I. Hedberg,
Function spaces and potential theory , Grundlehren der Mathematis-chen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag,Berlin, 1996.[2] L. Ambrosio, G. De Philippis, and L. Martinazzi,
Gamma-convergence of nonlocal perimeter func-tionals , Manuscripta Math. (2011), no. 3-4, 377–403.[3] L. Ambrosio, N. Fusco, and D. Pallara,
Functions of bounded variation and free discontinuity prob-lems , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York,2000.[4] E. Artin,
The Gamma function , Translated by Michael Butler. Athena Series: Selected Topics inMathematics, Holt, Rinehart and Winston, New York-Toronto-London, 1964.[5] J. Bergh and J. Löfström,
Interpolation spaces. An introduction , Springer-Verlag, Berlin-New York,1976.[6] J. Bourgain, H. Brezis, and P. Mironescu,
Another look at Sobolev spaces , Optimal control and partialdifferential equations, IOS, Amsterdam, 2001, pp. 439–455.[7] A. Braides, Γ -convergence for beginners , Oxford Lecture Series in Mathematics and its Applications,vol. 22, Oxford University Press, Oxford, 2002.[8] H. Brezis,
Functional analysis, Sobolev spaces and partial differential equations , Universitext,Springer, New York, 2011.[9] E. Brué, M. Calzi, G. E. Comi, and G. Stefani,
A distributional approach to fractional Sobolev spacesand fractional variation: asymptotics II , in preparation.[10] G. E. Comi and G. Stefani,
A distributional approach to fractional Sobolev spaces and fractionalvariation: existence of blow-up , J. Funct. Anal. (2019), no. 10, 3373–3435.[11] M. Cozzi and A. Figalli,
Regularity theory for local and nonlocal minimal surfaces: an overview ,Nonlocal and nonlinear diffusions and interactions: new methods and directions, Lecture Notes inMath., vol. 2186, Springer, Cham, 2017, pp. 117–158.[12] G. Dal Maso,
An introduction to Γ -convergence , Progress in Nonlinear Differential Equations andtheir Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. [13] J. Dávila, On an open question about functions of bounded variation , Calc. Var. Partial DifferentialEquations (2002), no. 4, 519–527.[14] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces , Bull.Sci. Math. (2012), no. 5, 521–573.[15] L. C. Evans and R. F. Gariepy,
Measure theory and fine properties of functions , Revised edition,Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.[16] H. Federer,
Geometric measure theory , Die Grundlehren der mathematischen Wissenschaften, Band153, Springer-Verlag New York Inc., New York, 1969.[17] L. Grafakos,
Classical Fourier analysis , 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer,New York, 2014.[18] ,
Modern Fourier analysis , 3rd ed., Graduate Texts in Mathematics, vol. 250, Springer, NewYork, 2014.[19] W. Gustin,
Boxing inequalities , J. Math. Mech. (1960), 229–239.[20] J. Horváth, On some composition formulas , Proc. Amer. Math. Soc. (1959), 433–437.[21] G. Leoni, A first course in Sobolev spaces , Graduate Studies in Mathematics, vol. 105, AmericanMathematical Society, Providence, RI, 2009.[22] F. Maggi,
Sets of finite perimeter and geometric variational problems , Cambridge Studies in Ad-vanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012.[23] V. Maz ′ ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerninglimiting embeddings of fractional Sobolev spaces , J. Funct. Anal. (2002), no. 2, 230–238.[24] ,
Erratum to: “On the Bourgain, Brezis and Mironescu theorem concerning limiting em-beddings of fractional Sobolev spaces” [J. Funct. Anal. (2002), no. 2, 230–238; MR1940355(2003j:46051)] , J. Funct. Anal. (2003), no. 1, 298–300.[25] T. Mengesha and D. Spector,
Localization of nonlocal gradients in various topologies , Calc. Var.Partial Differential Equations (2015), no. 1-2, 253–279.[26] M. Milman, Notes on limits of Sobolev spaces and the continuity of interpolation scales , Trans. Amer.Math. Soc. (2005), no. 9, 3425–3442.[27] A. C. Ponce,
Elliptic PDEs, measures and capacities , EMS Tracts in Mathematics, vol. 23, EuropeanMathematical Society (EMS), Zürich, 2016.[28] A. C. Ponce and D. Spector,
A boxing inequality for the fractional perimeter , Ann. Scuola Norm.Sup. Pisa Cl. Sci (5) (2017), to appear, available at https://arxiv.org/abs/1703.06195 .[29] A. Schikorra, D. Spector, and J. Van Schaftingen, An L -type estimate for Riesz potentials , Rev.Mat. Iberoam. (2017), no. 1, 291–303.[30] A. Schikorra, T.-T. Shieh, and D. Spector, L p theory for fractional gradient PDE with V M O coeffi-cients , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (2015), no. 4, 433–443.[31] A. Schikorra, T.-T. Shieh, and D. E. Spector, Regularity for a fractional p -Laplace equation , Commun.Contemp. Math. (2018), no. 1, 1750003, 6.[32] T.-T. Shieh and D. E. Spector, On a new class of fractional partial differential equations , Adv. Calc.Var. (2015), no. 4, 321–336.[33] , On a new class of fractional partial differential equations II , Adv. Calc. Var. (2018),no. 3, 289–307.[34] M. Šilhavý, Beyond fractional laplacean: fractional gradient and divergence (January 19, 2016),slides of the talk at the Department of Mathematics Roma Tor Vergata, available at https://doi.org/10.13140/RG.2.1.2554.0885 .[35] ,
Fractional vector analysis based on invariance requirements (Critique of coordinate ap-proaches) , M. Continuum Mech. Thermodyn. (2019), 1–22.[36] D. Spector,
An optimal Sobolev embedding for L (2018), preprint, available at https://arxiv.org/abs/1806.07588 .[37] , A noninequality for the fractional gradient (2019), preprint, available at https://arxiv.org/abs/1906.05541 .[38] E. M. Stein,
Singular integrals and differentiability properties of functions , Princeton MathematicalSeries, No. 30, Princeton University Press, Princeton, N.J., 1970.
DISTRIBUTIONAL APPROACH TO FRACTIONAL VARIATION: ASYMPTOTICS I 61 [39] ,
Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , PrincetonMathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993.[40] L. Tartar,
An introduction to Sobolev spaces and interpolation spaces , Lecture Notes of the UnioneMatematica Italiana, vol. 3, Springer, Berlin; UMI, Bologna, 2007.(G. E. Comi)
Fachbereich Mathematik, Universität Hamburg, Bundesstra ß e 55, 20146Hamburg, Germany E-mail address : [email protected] (G. Stefani) Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy
E-mail address ::