123 Onageneralizationof L -differentiability

DOI 10.1007/s00526-016-1004-9

Calculus of Variations

On a generalization of L

-differentiability

Daniel Spector¹

Received: 12 October 2015 / Accepted: 11 April 2016 / Published online: 20 May 2016

© Springer-Verlag Berlin Heidelberg 2016

Abstract In this paper we connect Calderón and Zygmund’s notion ofL^p-differentiability (Calderón and Zygmund, Proc Natl Acad Sci USA 46:1385–1389,1960) with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu (Optimal Control and Partial Differential Equations, pp.

439–455,2001). We show how the results of the former can be generalized to the setting of the latter, while the latter results can be strengthened in the spirit of the former. As a consequence of these results we give several new characterizations of Sobolev spaces, a novel condition for whether a function of bounded variation is in the Sobolev spaceW^1,1, and complete the proof of a characterization of the Sobolev spaces recently claimed in (Leoni and Spector, J Funct Anal 261:2926–2958,2011; Leoni and Spector, J Funct Anal 266:1106–1114,2014).

Mathematics Subject Classification 46E35 1 Introduction

1.1L^p-differentiability andL^p-Taylor approximations

L^p-differentiability was introduced by Calderón and Zygmund in their study of the local properties of solutions of elliptic differential equations [5,6]. It is a natural extension of classical differentiability in that it relaxes the requirement of the existence of a locally linear map in a uniform sense to its existence in an averaged sense. As the Sobolev spaces arise readily in the study of partial differential equations, it is not surprising that Sobolev functions possess anL^p-derivative. The following theorem asserting this fact was proven by Calderón and Zygmund [6, Theorem 12] (for a modern reference, see the monograph of Evans and Gariepy [8]).

Communicated by H. Brezis.

B

[email protected]

1 Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

Theorem 1.1 (Calderón and Zygmund)Suppose1≤ p<∞and f ∈W^1,p(R^N). Then 1

^pq

−

B(0,)|f(x+h)− f(x)− ∇f(x)·h|^pq dh ¹

pq →0 (1.1)

for_L^N almost every x ∈R^N, where1≤q ≤ _N^N_−p if1≤ p< N ,1≤q <∞if p= N , and1 ≤ q ≤ ∞if p > N (When q = ∞the left side of (1.1)is understood to be the L^∞(B(0, ))norm applied to the integrand).

While L^p-differentiability is a necessary condition for the inclusion of a function in W^1,p, the fact that it does not characterize the space is readily seen by the improvement of exponent in Theorem1.1. A natural question is then whether one can characterize the Sobolev spaces in the spirit of the condition (1.1). Several results have been given in this direction, for instance, a sort of converse to Theorem1.1due to Bagby and Ziemer [2], as well as some characterizations due to Swanson [12,13] that are based on the Calderón–Zygmund classes (see, for example, [16, Chapter 3, p. 132]). In fact, the ideas in [12] were virtually simultaneously introduced by Bourgain, Brezis, and Mironescu [3], who had shown that for

f ∈L^p(R^N)the simple condition lim sup

→0

R^N

−B(0,)

|f(x+h)− f(x)|^p

|h|^p dhd x<+∞

characterizesW^1,p(R^N)for 1 < p < +∞(which actually holds in the context of more general mollifiers, see Section1.2). The main idea in both [3,12] is that instead of considering the infinitesimal behavior of averaged difference quotients one should consider the integrated infinitesimal behavior—that is, one should utilize two integrals instead of one. Notice that the case p = 1 is excluded, since here one obtains a characterization not of W^1,1(R^N) but B V(R^N), the space of functions of bounded variation. To unify the approach for all 1≤p<+∞, let us introduce the following definition.

Definition 1.2 A functionf :R^N →Ris said to have a first orderL^p-Taylor approximation if f ∈L^p(R^N)and there exists a functionv∈L^p(R^N;R^N)such that

→0lim

R^N

−B(0,)

|f(x+h)− f(x)−v(x)·h|^p

|h|^p dhd x=0. (1.2)

The first result of this paper is the following theorem which asserts that functions in the Sobolev spaceW^1,p(R^N)possess a first orderL^p-Taylor approximation.

Theorem 1.3 Let1 ≤ p < ∞and f ∈ W^1,p(R^N). Then f has a first order L^p-Taylor approximation, and moreover, one has the stronger convergence

→0lim

R^N

−

B(0,)

|f(x+h)− f(x)− ∇f(x)·h|^pq

|h|^pq dh

¹

q

d x=0, (1.3) where1≤q≤ _N^N_−p if1≤p<N ,1≤q<∞if p=N , and1≤q≤ ∞if p>N .

What is quite surprising is that this property—the existence of a Taylor approximation in thisL^psense—in fact characterizes Sobolev functions. Precisely, we have the following theorem characterizing the Sobolev spaceW^1,p(R^N)in terms of theL^p(R^N)convergence (1.2).

Theorem 1.4 Let1 ≤ p <+∞. Then f ∈ W^1,p(R^N)if and only if f has a first order L^p-Taylor approximation.

The existence of anL¹-Taylor approximation is of a similar character to the assumption of Swanson in [13], since it allows one to gain some equi-integrability in an approximating sequence. If one does not make such a strong assumption, say that the limit (1.3) is finite but not zero, one obtains the following characterization of the space of functions of bounded variation.

Theorem 1.5 Suppose f ∈L¹(R^N). Then f ∈B V(R^N)if and only if there exists a function v∈L¹(R^N;R^N)such that

lim sup

→0

R^N

−

B(0,)

|f(x+h)− f(x)−v(x)·h|^q

|h|^q dh ¹

q

d x<+∞ (1.4) for any1≤q≤ _N^N₋₁.

In general, the limit (1.4) is bounded by a constant times|D^sf|(R^N), the total variation of the singular portion of the measureD f. However, whenq=1 one can explicitly compute the limit, which is given by

Theorem 1.6 Suppose f ∈B V(R^N). Then writing D f=vL^N+D^sf wherev∈L¹(R^N;R^N) and D^sf ⊥L^N, one has

→lim0

R^N

−

B(0,)

|f(x+h)− f(x)−v(x)·h|

|h| dhd x=K_1,N|D^sf|(R^N).

HereK_1,N is as below in Sect.1.2. These results have been announced in [11] (with the exception of Theorem1.6) with proofs of several implications, and it is our intention here to provide the complete proofs of all of the claims. However, another major point we wish to address is the fact that both Theorem1.1and Theorem1.3hold in the context of a larger framework—the use of more general approximations of the identity as seen in the work of Bourgain, Brezis, and Mironescu [3]. We now develop the necessary background for these results and several later in the paper.

1.2 Connections to the work of Bourgain, Brezis, and Mironescu

Let us suppose that⊂R^N is open, bounded, and smooth (or all ofR^N), and that{ρ} ⊂ L¹(R^N)are radial mollifiers that satisfy

ρ≥0,

R^Nρ(x)d x=1, (1.5)

→lim0

|x|>δρ(x)d x=0 for allδ >0. (1.6) Takingρ(x) = _|B(0,)|¹ χB(0,)(x), we revert to the assumptions of the previous section.

However, one has the possibility of new interesting examples, such as that of the “Gagliardo semi-norms”:

ρ(x)=c χB(0,R)

|x|^N+(−1)r, forc→0 as→0, or more generally (cf. Brezis [4])

ρ(x)=

⎧⎨

⎩

0 if 0<|x| ≤, aψ(|x|) if <|x| ≤1, 0 if 1<|x|,

whereψ∈L¹_loc(0,1)satisfies ₁

0 ψ(r)r^N−1dr= +∞

and

a:=

₁

ψ(r)r^N−1dr −1

→0 as→0.

Then the work¹of Bourgain, Brezis, and Mironescu [3] showed two results of particular relevance here. Firstly, if 1≤p<+∞and f ∈C²()then one has

→lim0

|f(x)− f(y)|^p

|x−y|^p ρ(x−y)d y=K_p,N|∇f(x)|^p (1.7) for everyx∈(see [3], p. 444) for

K_p,N :=

−

S^N−1|e₁·σ|^pdH^N−1(σ ).

Secondly, they showed that if 1< p<+∞and f ∈L^p(), one has f ∈W^1,p()if and only if

lim sup

→0

|f(x)− f(y)|^p

|x−y|^p ρ(x−y)d yd x <+∞, and in that case

→0lim

|f(x)− f(y)|^p

|x−y|^p ρ(x−y)d yd x =Kp,N

|∇f(x)|^p d x. (1.8) In fact, whenρ(x)= _|B(0,)|¹ χB(0,)(x)one can deduce the convergence (1.7) (even for W^1,pfunctions, compare with Corollary1.8below) as a consequence ofL^p-differentiability (in an improved form allowing for_|h¹_|p in place of¹p, cf. Ambrosio, Fusco, and Pallara [1]), since the estimate

|A^p−B^p| ≤ |A−B|C(1+A^p−1+B^p−1), (1.9) and Hölder’s inequality imply

→lim0

−

B(x,)

|f(y)− f(x)|^p

|x−y|^p d y−K_p,N|∇f(x)|^p

= lim

→0

−

B(x,)

|f(y)− f(x)|^p

|x−y|^p d y−

−

B(x,)

∇f(x)· y−x

|y−x| ^p d y

≤lim

→0

−

B(x,)

|f(y)− f(x)− ∇f(x)·(y−x)|^p

|x−y|^p d y ¹

p

×

−

B(x,)C(1+|f(y)− f(x)|^p

|x−y|^p + |∇f(x)|^p)d y ¹

p

,

1We follow the convention of the subsequent paper of Ponce [14] in taking mollifiers indexed byinstead ofn.

while the difference quotient appearing in the second quantity can be bounded by observing that

−

B(x,)

|f(y)− f(x)|^p

|x−y|^p d y ¹

p

≤

−

B(x,)

|f(y)− f(x)− ∇f(x)·(y−x)|^p

|x−y|^p d y ¹

p

+

−B(x,)

∇f(x)· y−x

|y−x|

^p d y ¹

p.

The same computation follows for general mollifiers, and in this spirit, one might ask whether the stronger pointwise convergence result of Theorem1.1holds in this setting. Indeed, we have the following theorem to this effect.

Theorem 1.7 Let1≤ p <∞and f ∈W_loc^1,p(R^N). Supposeρare radial non-increasing and satisfy(1.5)and(1.6). Then for anyη >0one has

→0lim

B(0,η)

|f(x+h)− f(x)− ∇f(x)·h|^pq

|h|^pq ρ(h)dh=0

forL^Nalmost every x ∈R^N, where1≤q≤ _N^N_−p if1≤p<N and1≤q<∞if p≥N . Combined with the analysis preceding the theorem, one has the following corollary, a version of the pointwise convergence (1.7) for functions inW^1,p.

Corollary 1.8 Let1≤ p<∞and f ∈W^1,p(). Supposeρ are radial non-increasing and satisfy(1.5)and(1.6). Then forL^Nalmost every x ∈R^N one has

→0lim

B(0,η)

|f(x+h)− f(x)|^p

|h|^p ρ(h)dh=K_p,N|∇f(x)|^p for everyη >0such that B(x, η)⊂.

This observation prompts us to return to the energy convergence (1.8) and perform a similar estimate. In this setting, one observes that using Minkowski’s inequality for integrals implies

→lim0

|f(y)− f(x)|^pq

|x−y|^pq ρ(x−y)d y ¹

q −K

1 q

pq,N|∇f(x)|^p

d x

≤ lim

→0

⎛

⎝

|f(y)− f(x)|^p

|x−y|^p −

∇f(x)· y−x

|x−y| ^pq

ρ(x−y)d y ¹_q⎞

⎠d x,

which using again the inequality (1.9) and Hölder’s inequality yields

→0lim

|f(y)− f(x)|^pq

|x−y|^pq ρ(x−y)d y ¹

q −K

1

pq,Nq |∇f(x)|^p

d x

≤lim

→0

|f(y)− f(x)− ∇f(x)·(y−x)|^pq

|x−y|^pq ρ(x−y)d y ¹

q

d x ¹_p

×

C(1+|f(y)− f(x)|^pq

|x−y|^pq + |∇f(x)|^pq) ρ(x−y)d y ¹

q

d x _p¹

. The same estimate as before shows that the quantities in question are finite, provided one can establish a generalization of the estimate (1.3) to other families of mollifiers. The following theorem shows that for p >1 and pq< p^∗, the Sobolev critical exponent (see Sect.2), one has such a generalization.

Theorem 1.9 Let1< p<∞and f ∈W_loc^1,p(R^N). Supposeρare radial non-increasing and satisfy(1.5)and(1.6). Then for anyη >0one has

→0lim

K

B(0,η)

|f(x+h)− f(x)− ∇f(x)·h|^pq

|h|^pq ρ(h)dh ¹_q

d x=0 (1.10) for all K ⊂R^N compact, and where1≤q< _N−^N_p if1<p<N .

The degeneracy when p = 1 is more than an issue withW^1,1 versus B V, the space of functions of bounded variation. Whereas previously in the paper [10], the author and G.

Leoni had given an example of a function f ∈B V(0,1)showing one could not expect such a convergence for f ∈B V, we present a counterexample to theW^1,1case in Sect.5(which is an adaptation a counterexample of Ponce presented by Nguyen in [15]) of a family of radial non-increasing mollifiersρand a function f ∈W^1,1(0,1)for which

→lim0

₁

0

₁

0

|f(x)− f(y)|^q

|x−y|^q ρ(x−y)d y ¹_q

d x= +∞

for anyq>1. In particular, this shows the restrictionp>1 in Theorem1.9is necessary. It would be interesting to understand whether for 1< p<Nthe assumption thatpq< p^∗is also necessary or an artifact of the proof.

The plan of the paper is as follows. In Sect.2we review our notation and recall some lemmata regarding weakly differentiable functions. In Sect.3we restrict our attention to mollification by the characteristic function, completing the proofs of Theorems1.3,1.4, and 1.5announced in [11]. We also prove Theorem1.6. In Sect.4we treat the case of general radial non-increasing mollifiers and prove the main results from Sect.1.2. Finally, in Sect.5we treat the case of non-smooth domains in relation to the works [9,10]. Here we substantiate a claim in the paper [9], provided one is willing restrict oneself to radial non-increasing mollifiers (or mollifiers that admit a family of radial non-increasing majorants—see Sect.5).

2 Preliminaries

In what follows for 1≤ p<N we denote by p^∗= N p

N−p

the Sobolev critical exponent and formally takep^∗= +∞ifp>N. We use the notation

−B(x,r)f(y)d y:= 1

|B(x,r)|

B(x,r) f(y)d y

to denote the average value of the function f over the open ball centered atxand of radius r, which we sometimes will shorten to

−f, when the set of integration is implied by the context.

We utilizeS^N−1for the unit sphere inR^NandH^N−1for the(N−1)-dimensional Hausdorff measure.

For f ∈ W_loc^1,p(R^N)we write∇f for the distributional gradient of f, while for f ∈ B V_loc(R^N)we writeD f. Let us recall thatD f admits a decompositionD f = ∇fL^N+D^sf, where we utilize with an abuse of notation ∇f ∈ L¹_loc(R^N;R^N) to denote the Radon–

Nikodym derivative of D f with respect to the Lebesgue measure and D^sf its singular measure (D^sf ⊥L^N).

We utilize the notation

M(g)(x):=sup

t>0

−B(x,t)|g(y)|d y,

to denote the Hardy–Littlewood maximal function.

We denote byC a constant which may depend on the dimension N and the parameters p,q, though whose precise value may change from line to line.

Let us here record some inequalities that will be useful in the sequel. The first lemma is a Sobolev–Poincaré inequality for Sobolev functions on balls, a proof of which can be found as Theorem 2 on page 141 of [8].

Lemma 2.1 Let1≤p<N . Then there exists a constant C=C(N,p) >0such that

−B(x,r)|f(y)−

−f|^p∗ d y ¹

p∗

≤Cr

−B(x,r)|∇f(y)|^pd y ¹

p ,

for all B(x,r)⊂R^N and f ∈W^1,p(B(x,r)).

Essentially the same proof gives the following analogous result for functions of bounded variation (which is Theorem 1 on page 189 of [8]).

Lemma 2.2 There exists a constant C=C(N) >0such that

−B(x,r)|f(y)−

−f|^1∗d y ¹

1∗

≤Cr|D f|(B(x,r))

r^N .

for all B(x,r)⊂R^N and f ∈B Vloc(R^N).

We prove the following

Lemma 2.3 Suppose f ∈W_loc^1,p(R^N)for some1≤p<∞, and that1≤q<+∞is such that pq≤p^∗. Then there exists a C=C(p,q,N) >0such that for all t>0

1 t^N+pq

B(0,t)|f(x+h)− f(x)− ∇f(x)·h|^pq dh

≤C

−B(0,t)|∇f(x+h)− ∇f(x)|^s dh ^pq

s

+C 1

t

−B(0,t)|f(x+z)− f(x)− ∇f(x)·z|d z pq

, where1≤s≤p is such that s^∗=pq.

Proof We begin with the estimate 1

t^N+pq

B(0,t)|f(x+h)− f(x)− ∇f(x)·h|^pqdh

≤ C t^N+pq

B(0,t)|f(x+h)−

−f − ∇f(x)·h|^pqdh

+ C t^N+pq

B(0,t)|

−

B(0,t)f(x+z)− f(x)− ∇f(x)·z d z|^pq dh where we have added and subtracted

−f(x+z)d z(where the integral is taken overB(0,t)) and used the fact that

∇f(x)·z d z=0. The second term agrees with what we aim to show in this lemma, so it remains to show that

1 t^N+pq

B(0,t)|f(x+h)−

−f − ∇f(x)·h|^pqdh

≤C

−B(0,t)|∇f(x+h)− ∇f(x)|^s dh ^pq

s

and the result is demonstrated. However, utilizing the fact that 1 ≤ s ≤ p, we have that f ∈W_loc^1,s(R^N), and so applying Lemma2.1to

v(h):= f(x+h)−

−f − ∇f(x)·h

(note that f ∈W_loc^1,s(R^N)impliesv∈W^1,s(B(0,R))for every 0<R<∞) we obtain

B(0,t)|f(x+h)−

−f − ∇f(x)·h|^pq dh

≤C

B(0,t)|∇f(x+h)− ∇f(x)|^s dh ^pq

s ,

since

−v=0 and∇v(h)= ∇f(x+h)− ∇f(x). Then dividing byt^N+pq, we obtain the

desired inequality.

Analogously we have

Lemma 2.4 Suppose f ∈B V_loc(R^N)that1≤q≤1^∗. Then there exists a C=C(q,N) >

0such that for all t>0 1 t^N+q

B(0,t)|f(x+h)− f(x)− ∇f(x)·h|^q dh

≤C

−B(0,t)|∇f(x+h)− ∇f(x)|dh+ |D^sf|(B(0,t)) q

+C 1

t

−B(0,t)|f(x+z)− f(x)− ∇f(x)·z|d z q

.

Proof As before 1 t^N+q

B(0,t)|f(x+h)− f(x)− ∇f(x)·h|^q dh

≤ C t^N+q

B(0,t)|f(x+h)−

−f − ∇f(x)·h|^q dh

+ C t^N+q

B(0,t)|

−B(0,t)f(x+z)− f(x)− ∇f(x)·z d z|^q dh,

though here instead we have the estimate 1

t^N+q

B(0,t)|f(x+h)−

−f − ∇f(x)·h|^q dh

= C t^q

−B(0,t)|f(x+h)−

−f − ∇f(x)·h|^qdh

≤ C t^q

−B(0,t)|f(x+h)−

−f − ∇f(x)·h|^1∗dh ^q

1∗

= C

t^1∗

−B(0,t)|f(x+h)−

−f − ∇f(x)·h|^1∗ dh ^q

1∗

. Now, applying Lemma2.2tov(h):= f(x+h)−

−f − ∇f(x)·h, we have C

t^1∗

−B(0,t)|f(x+h)−

−f − ∇f(x)·h|^1∗dh ¹

1∗

≤C

−B(0,t)|∇f(x+h)− ∇f(x)|d x+ |D^sf|(B(0,t)),

and this implies the desired result.

Finally, whenp> N we have Morrey’s inequality for Sobolev functions with a precise estimate for the exponent of Hölder continuity as given in the following Lemma (which is stated and proved as Theorem 3 on page 189 of [8]).

Lemma 2.5 Let N <p<+∞. There exists a C=C(p,N) >0such that

|f(z)− f(y)| ≤Cr

−B(x,r)|∇f(w)|^pdw ¹_p

for all B(x,r)⊂R^N, f ∈W^1,p(B(x,r)), andL^N almost every y,z∈B(x,r).

3 L^p-Taylor approximations andW^1,p

Let us recall that in [11] we gave a proof of Theorem1.3in the regime 1≤p<N, Theorem 1.4and the necessity of f ∈ B V(R^N)supposing that the lim sup is bounded in Theorem 1.5. Therefore we here complete the arguments to Theorem1.3in the regime p≥ N and Theorem1.5. We also prove Theorem1.6.

We begin with the proof of Theorem1.3in the regimep≥N.

Proof Letp ≥ N and f ∈ W^1,p(R^N). Ifq <+∞, then the result follows by exactly the same argument in [11] observing that although Lemma2.3was only stated for p < N in [11], we here proved it for all 1≤p<+∞and any 1≤q<+∞. Thus it remains to treat the caseq= +∞. The estimate for Morrey’s inequality given in Lemma2.5implies that for anyN <s< pone has

|f(x+h)− f(x)− ∇f(x)·h|^s

|h|^s ≤C

−B(x,|h|)|∇f(z)− ∇f(x)|^sd z

and so sup

h∈B(0,)

|f(x+h)− f(x)− ∇f(x)·h|^p

|h|^p ≤C sup

h∈B(0,)

−B(x,|h|)|∇f(z)− ∇f(x)|^s d z ^p_s

.

Now, the right hand side tends to zero pointwise as→0, while sup

h∈B(0,)

−

B(x,|h|)|∇f(z)− ∇f(x)|^s d z ^p

s ≤(M(|∇f|^s)(x))^ps + |∇f(x)|^p,

and so the result follows from Lebesgue’s dominated convergence theorem and the bound- edness ofM:L^r(R^N)→L^r(R^N)for all 1<r≤ +∞

We now give a proof of Theorem1.5, arguing as in [11], mutatis mutandis.

Proof As argued in [11], the finiteness of the limit implies f ∈ B V(R^N), so it suffices to show that for f ∈B V(R^N)we can find a bound for the limit.

For any 0< <1, we expand the integrand on concentric rings

−B(0,)

|f(x+h)− f(x)− ∇f(x)·h|^q

|h|^q dh

= ∞ i=0

1 ^N|B(0,1)|

B(0, 2i)\B(0,

2i+1)

|f(x+h)− f(x)− ∇f(x)·h|^q

|h|^q dh.

We now make estimates fori∈Nfixed. We have 1

^N

B(0, 2i)\B(0,

2i+1)

|f(x+h)− f(x)− ∇f(x)·h|^q

|h|^q dh

≤ 1 ^N

2ⁱ⁺¹

₋q

B(0,_2i)\B(0,_2i+1 )|f(x+h)− f(x)− ∇f(x)·h|^q dh

≤ 2^q 2^{i N}

2ⁱ

₋N−q

B(0,

2i)|f(x+h)− f(x)− ∇f(x)·h|^q dh.

Here, in place of Lemma 2.3we utilize Lemma2.4to deduce that 2ⁱ

_−N−q

B(0,

2i)|f(x+h)− f(x)− ∇f(x)·h|^q dh

≤C −

B(0,

2i)|∇f(x+h)− ∇f(x)|^p dh+ 2ⁱ

N

|D^sf|(B(x, /2ⁱ)) q

+C

1 0

−

B(0,t

2i)|∇f(x+t z)− ∇f(x)|d z+ 2ⁱ

t N

|D^sf|(B(x,t/2ⁱ))dt q

.

Here we have used the estimate

−B(0,

2i)|f(x+z)− f(x)− ∇f(x)·z| d z

≤ ₁

0

−

B(0,t

2i)|∇f(x+t z)− ∇f(x)|d z+ + 2ⁱ

t N

|D^sf|(B(x,t/2ⁱ))dt for the second term of Lemma2.4, whose argument is almost identical to the one used in the upper bound in the proof of Theorem1.6(and is originally due to Ponce [14], p. 248).

Therefore, summing iniand applying the basic inequality

i|ai|_q¹

≤

i|ai|^q1 (which follows from subadditivity of the functions→s^1q), we have

−B(0,)

|f(x+h)− f(x)− ∇f(x)·h|^q

|h|^q dh

¹_q

≤C ∞ i=0

1 2ⁱ

_N/q

−B(0,

2i)|∇f(x+h)− ∇f(x)|dh+ 2ⁱ

N

|D^sf|(B(x, /2ⁱ)) +C

∞ i=0

1 2ⁱ

_{N/q 1}

0

−B(0,t

2i)|∇f(x+t z)− ∇f(x)|d z +

2ⁱ t

N

|D^sf|(B(x,t/2ⁱ))dt.

Integrating the preceding inequality overx ∈R^N and making use of Tonelli’s theorem we obtain

R^N

−B(0,)

|f(x+h)− f(x)− ∇f(x)·h|^q

|h|^q dh

¹

q

d x

≤C ∞ i=0

1 2ⁱ

_N/q

−B(0, 2i)

R^N|∇f(x+h)− ∇f(x)|d xdh +

2ⁱ

N

|D^sf|(B(x, /2ⁱ))d x

+C ∞ i=0

1 2ⁱ

_{N/q 1}

0

−B(0,t 2i)

R^N|∇f(x+t z)− ∇f(x)|d z +

R^N

2ⁱ t

N

|D^sf|(B(x,t/2ⁱ))d x

dt.

However, ifh,z∈B(0, )andt∈(0,1)we have max

R^N|∇f(x+h)− ∇f(x)|d x,

R^N|∇f(x+t z)− ∇f(x)|d x

≤ sup

w∈B(0,)

R^N|∇f(x+w)− ∇f(x)|d x,

and observe that this bound is independent ofi∈N. Thus,

R^N

−

B(0,)

|f(x+h)− f(x)− ∇f(x)·h|^q

|h|^q dh

¹

q

d x

≤C ∞ i=0

1 2ⁱ

N/q

sup

w∈B(0,)

R^N|∇f(x+w)− ∇f(x)|d x +C

∞ i=0

1 2ⁱ

_N/q

R^N

2ⁱ

N

|D^sf|(B(x, /2ⁱ))d x

+ ₁

0

R^N

2ⁱ t

N

|D^sf|(B(x,t/2ⁱ))d xdt

. Finally we observe that

₁

0

R^N

2ⁱ t

N

|D^sf|(B(x,t/2ⁱ))d xdt

= ₁

0

2ⁱ t

N

B(0,t 2i)

R^N d|D^sf|(x−y)d ydt

= |D^sf|(R^N), while a similar argument shows

R^N

2ⁱ

N

|D^sf|(B(x, /2ⁱ))d x= |D^sf|(R^N), and these equalities hold for alli∈N.

Thus, as the infinite series is summable, sending→0 and using continuity of translation inL¹(R^N)we obtain

→0lim

R^N

−B(0,)

|f(x+h)− f(x)− ∇f(x)·h|^q

|h|^q dh

¹

q

d x≤C|D^sf|(R^N).

We now calculate the explicit limit of theL¹-Taylor approximation for a function f ∈ B V(R^N), proving the claim of Theorem1.6. Observe here that the fact thatq=1 means we do not need to use the Sobolev embedding theorem and therefore the computation is cleaner.

Proof We first argue the upper bound lim sup

→0

R^N

−B(0,)

|f(x+h)− f(x)− ∇f(x)·h|

|h| dhd x≤K_1,N|D^sf|(R^N) (3.1) Fixψ ∈C_c^∞(B(0,1))radial, non-negative and

ψ=1 and defineψδ(x):=_δ¹Nψ(^x_δ). For f ∈B V(R^N)we denote by f_δthe convolution f ∗ψδ. Then ifD f is the measure derivative of f, one hasD f_δ=(∇f)δ+(D^sf)δ, and so the triangle inequality and Tonelli’s theorem imply

R^N

−B(0,)

|f_δ(x+h)− f_δ(x)−(∇f)_δ(x)·h|

|h| dhd x

≤ ₁

0

R^N

−B(0,)|(∇f_δ(x+sh)−(∇f)δ(x))· h

|h||dhd xds

≤ ₁

0

R^N

−B(0,)|(∇f)δ(x+sh)−(∇f)δ(x)|dhd xds

+ ₁

0

R^N

−

B(0,)

(D^sf)δ(x+sh)· h

|h|

dhd xds

= ₁

0

−

B(0,)

R^N|(∇f)δ(x+sh)− ∇f_δ(x)|dhd xds +

−B(0,)

R^N

(D^sf)δ(y)· h

|h|

d ydh. Now

−

B(0,)

R^N

(D^sf)δ(y)· h

|h| d ydh

= 1

^N|B(0,1)|

0

∂B(0,t)

R^N

(D^sf)_δ(y)·σ d ydH^N−1(σ )dt

= 1

^N|B(0,1)|

0

∂B(0,t)

R^N|(D^sf)δ(y)·σ|d ydH^N−1(σ )dt

= 1

^N|B(0,1)|

0

t^N−1dt

S^N−1

R^N|(D^sf)_δ(y)·σ|d ydH^N−1(σ )

=K1,N|(D^sf)_δ|(R^N) for

K_1,N =

−

S^N−1|e₁·σ|dH^N−1(σ )

as in Sect.1.2. Therefore sendingδ →0⁺, Fatou’s lemma, Lebesgue’s dominated conver- gence theorem, the strict convergence of the measures(D^sf)_δ→D^sf yields

R^N

−B(0,)

|f(x+h)− f(x)− ∇f(x)·h|

|h| dhd x

≤ ₁

0

−B(0,)

R^N|∇f(x+sh)− ∇f(x)|d xdhds+K_1,N|D^sf|(R^N) Now taking the limsup as →0 the first term vanishes as argued in the previous theorems (by continuity of translation onL¹(R^N)), and we thus obtain (3.1).

To prove the lower bound, we simply estimate

R^N

−B(0,)

|f(x+h)− f(x)|

|h| dhd x−K1,N

R^N|∇f(y)|d y

=

R^N

−

B(0,)

|f(x+h)− f(x)|

|h| dhd x−

−

B(0,)

R^N

∇f(y)· h

|h| d ydh

≤

R^N

−B(0,)

|f(x+h)− f(x)− ∇f(x)·h|

|h| dhd x,

and taking the liminf as→0, an application of the result of Dávila [7] that

→lim0

R^N

−

B(0,)

|f(x+h)− f(x)|

|h| dhd x=K_1,N|D f|(R^N), and the decomposition

|D f|(R^N)= |D^sf|(R^N)+

R^N|∇f(y)|d y

yields the desired lower bound.

4 General mollifiers

We will here give proofs of the results asserted in Sect.1.2. We begin with Theorem1.7.

Proof of Theorem 1.7 We will show that

→0lim

B(0,η)

|f(x+h)− f(x)− ∇f(x)·h|^pq

|h|^pq ρ(h)dh ¹

pq =0

forp-Lebesgue points of∇f, and henceL^N almost everyx ∈R^N. Let us first recall that as a monotone decreasing function of|h|,ρhas at most countably many jump discontinuities, and possesses inner and outer limits for every value of|h|, and therefore its suffices to prove the result in the case whereρis inner continuous (which always dominatesρ).

In order to apply Lemma2.3, we expand the desired quantity on concentric rings to obtain

B(0,η)

|f(x+h)− f(x)− ∇f(x)·h|^pq

|h|^pq ρ(h)dh

= ∞ i=0

B(0,₂^η_i)\B(0,_2i^η₊₁)

|f(x+h)− f(x)− ∇f(x)·h|^pq

|h|^pq ρ(h)dh

≤ ∞ i=0

ρ

η 2ⁱ⁺¹

η 2ⁱ⁺¹

₋pq

B(0,₂^η_i)|f(x+h)− f(x)− ∇f(x)·h|^pq dh

=2^pq+2N ∞ i=0

ρ η 2ⁱ⁺¹

η 2ⁱ⁺²

Nη 2ⁱ

_−pq−N

×

B(0,₂^η_i)|f(x+h)− f(x)− ∇f(x)·h|^pqdh.

Then applying the estimate from Lemma2.3, we have

B(0,η)

|f(x+h)− f(x)− ∇f(x)·h|^pq

|h|^pq ρ(h)dh (4.1)

≤C ∞ i=0

ρ

η 2ⁱ⁺¹

η 2ⁱ⁺²

N

max

F₁ⁱ(x),F₂ⁱ(x)

, (4.2)

where

F₁ⁱ(x):= −

B(0,^η

2i)|∇f(x+h)− ∇f(x)|^s dh ^pq

s