PROXIMAL SUBDIFFERENTIABLE FUNCTIONS SATISFY A LUSIN
PROPERTY OF CLASS C 2
arXiv:1706.07980v1 [math.FA] 24 Jun 2017
D. AZAGRA, J. FERRERA, M. GARCÍA-BRAVO, AND J. GÓMEZ-GIL
Abstract. Let f : Rn → R be a function. Assume that for a measurable set Ω and almost
every x ∈ Ω there exists a vector ξx ∈ Rn such that
f (x + h) − f (x) − hξx , hi
lim inf
> −∞.
h→0
|h|2
Then we show that f satisfies a Lusin-type property of order 2 in Ω, that is to say, for every
ε > 0 there exists a function g ∈ C 2 (Rn ) such that Ln ({x ∈ Ω : f (x) 6= g(x)}) ≤ ε. In particular
every function which has a nonempty proximal subdifferential almost everywhere also has the
Lusin property of class C 2 . We also obtain a similar result (replacing C 2 with C 1 ) for the
Fréchet subdifferential. Finally we provide some examples showing that this kind of results are
no longer true for Taylor subexpansions of higher order.
A classical theorem of Lusin states that for every Lebesgue measurable function f : Rn → R
and every ε > 0 there exists a continuous function g : Rn → R such that
(1)
Ln ({x ∈ Rn : f (x) 6= g(x)}) ≤ ε.
Here, as in the rest of this note, Ln denotes the Lebesgue outer measure in Rn .
Several authors have shown that one can take g of class C k , provided that f has some regularity
properties of order k (for instance, locally bounded distributional derivatives up to the order
k, or Taylor expansions of order k almost everywhere). If, given a differentiability class C and
a function f : Rn → R we can find, for each ε > 0, a function g ∈ C satisfying (1), we will say
that f has the Lusin property of class C.
The first of such results was discovered by Federer [11, p. 442], who showed that a.e differentiable functions (and in particular locally Lipschitz functions) have the Lusin property of
class C 1 . H. Whitney [21] improved this result by showing that a function f : Rn → R has
approximate partial derivatives of first order a.e. if and only if f has the Lusin property of
class C 1 .
In [8, Theorem 13] Calderon and Zygmund established analogous results of order k for the
classes of Sobolev functions W k,p(Rn ). Other authors, including Liu [17], Bagby, Michael and
Ziemer [4, 19], Bojarski, Hajlasz and Strzelecki [5, 6], and Bourgain, Korobkov and Kristensen
[7] have improved Calderon and Zygmund’s result in different ways, by obtaining additional
estimates for f − g in the Sobolev norms, as well as the Bessel capacities or the Hausdorff
contents of the exceptional sets where f 6= g. In [7] some Lusin properties of the class BVk (Rn )
(of integrable functions whose distributional derivatives of order up to k are Radon measures)
are also established.
Date: June 5, 2017.
Key words and phrases. Lusin property of order 2, Proximal subdifferential.
1
2
D. AZAGRA, J. FERRERA, M. GARCÍA-BRAVO, AND J. GÓMEZ-GIL
For the special class of convex functions f : Rn → R, Alberti and Imonkulov [2, 15] showed
that every convex function has the Lusin property of class C 2 (with g not necessarily convex
in (1)). More recently Azagra and Hajlasz [3] have proved that g can be taken to be C 1
and convex in (1) if and only if either f is essentially coercive (meaning that f is coercive
up to a linear perturbation) or else f is already C 1 (in which case taking g = f is the only
possible option); they have also shown that if f : Rn → R is strongly convex then for every
A ⊂ Rn of finite measure and every ε > 0 there exists g : Rn → R convex and C 1,1 such that
Ln ({x ∈ A : f (x) 6= g(x)}) ≤ ε.
On the other hand, generalizing Whitney’s result [21] to higher orders of differentiability, Liu
and Tai established in [18] that a function f : Rn → R has the Lusin property of class C k if
and only if f is approximately differentiable of order k almost everywhere (and if and only if f
has an approximate (k − 1)-Taylor polynomial at almost every point).
In this note we will answer the following question (which we think may be quite natural for
people working on nonsmooth analysis or viscosity solutions to PDE such as Hamilton-Jacobi
equations): do functions with nonempty subdifferentials a.e. have Lusin properties of order C 1
or C 2 ? By subdifferentials we mean the Fréchet subdifferential, or the proximal subdifferential, or the second order viscosity subdifferential; see [9, 10, 12] and the references therein for
information about subdifferentials and their applications. As we will see the answer is positive:
Fréchet subdifferentiable functions have the Lusin property of class C 1 , and functions with
nonempty proximal subdifferentials a.e. (in particular functions with a.e. nonempty viscosity
subdifferentials of order 2) have the Lusin property of class C 2 .
This question can be formulated in a more general form (perhaps appealing to a wider audience)
as a problem about Taylor subexpansions: given k ∈ N and a function f : Rn → R, assume
that for almost every x ∈ Rn there exists a polynomial Px of degree less than or equal to k − 1
such that
f (y) − Px (y)
lim inf
> −∞.
y→x
|y − x|k
Is then true that f has the Lusin property of order k?
The results of this note will show that the answer to this question is positive for k = 1, 2, but
negative for k ≥ 3.
In the case k = 1 the proof is very simple and natural.
Theorem 1. Let Ω ⊂ Rn be a measurable set, and f : Ω → R a function. Assume that for
almost every x ∈ Ω we have
(2)
lim inf
y→x
f (y) − f (x)
> −∞.
|y − x|
Then, for every ε > 0 there exists a function g ∈ C 1 (Rn ) such that
Ln ({x ∈ Ω : f (x) 6= g(x)}) ≤ ε.
Proof. Assume for the time being that Ω is bounded. Let us call N ⊂ Ω the set of points for
which (2) does not hold; then Ln (N) = 0. By replacing Ω with Ω \ N we may and do assume
in what follows that N = ∅. By the regularity of the measure Ln we may also assume that Ω
is compact. Note that the assumption (2) implies that f is lower semicontinuous on Ω, and in
PROXIMAL SUBDIFFERENTIABLE FUNCTIONS SATISFY A LUSIN PROPERTY OF CLASS C 2
3
particular f is measurable. Define for each j ∈ N,
1
Ej := x ∈ Ω : f (y) − f (x) ≥ −j|y − x| for all y ∈ B x,
∩ Ω ∩ {x ∈ Ω : |f (x)| ≤ j} .
j
Because f is lower semicontinuous the sets
1
x ∈ Ω : f (y) − f (x) ≥ −j|y − x| for all y ∈ B x,
∩Ω
j
are closed, and using the measurability of f this implies that each set Ej is measurable. These
sets form an increasing sequence such that
∞
[
Ω=
Ej ,
j=1
so we have
lim Ln (Ω \ Ej ) = 0,
j→∞
and therefore, for a given ε > 0 we may find j0 ∈ N large enough such that Ln (Ω \ Ej0 ) < 2ε .
Take now x, y ∈ Ej0 . If |y − x| ≤ j10 then we have
|f (y) − f (x)| ≤ j0 |y − x| and |f (x)| ≤ j0 .
On the other hand, if x, y ∈ Ej0 and |y − x| > 1/j0 then we trivially get
|f (y) − f (x)| ≤ 2 sup |f (z)| ≤ M0 |y − x|,
z∈Ω
where M0 := 2j0 (1 + supz∈Ω |f (z)|). In either case we see that there exists M0 such that
|f (y) − f (x)| ≤ M0 |y − x| and |f (x)| ≤ M0 , for all x, y ∈ Ej0 .
That is, f is bounded and M0 -Lipschitz on Ej0 . Then we can extend f to a Lipschitz function
F on Rn , for instance by using the McShane-Whitney formula
F (x) = inf {f (y) + M0 |x − y|},
y∈Ej0
which defines an M0 -Lipschitz function on Rn that coincides with f on Ej0 . Obviously we have
ε
Ln ({x ∈ Ω : f (x) 6= F (x)}) ≤ Ln (Ω \ Ej0 ) < .
2
But according to the result of Federer’s that we mentioned above (see also [13, Theorem 6.11]),
Lipschitz functions have the C 1 Lusin property, so we may find another function g ∈ C 1 (Rn )
such that Ln ({x ∈ Ω : F (x) 6= g(x)}) < 2ε . Thus we conclude that
Ln ({x ∈ Ω : f (x) 6= g(x)}) = Ln ({x ∈ Ej0 : F (x) 6= g(x)} ∪ {x ∈ Ω \ Ej0 : f (x) 6= g(x)}) ≤
ε ε
≤ Ln ({x ∈ Ej0 : F (x) 6= g(x)}) + Ln (Ω \ Ej0 ) ≤ + = ε.
2 2
Now let us consider the general case that f is not necessarily bounded. We can write
∞
[
Ω=
Ωj , where Ωj := Ω ∩ B(0, j).
j=1
4
D. AZAGRA, J. FERRERA, M. GARCÍA-BRAVO, AND J. GÓMEZ-GIL
According to the previous argument, for each j ∈ N there exists a function gj ∈ C 1 (Rn ) such
that
ε
Ln ({x ∈ Ωj+2 : gj (x) 6= f (x)}) ≤ j .
6
∞
Let (ψj )∞
be
a
C
smooth
partition
of
unity
subordinated
to the covering {B(0, j + 1) \
j=1
∞
n
B(0, j − 1)}j=1 of R , and let us define
g(x) =
∞
X
ψj (x)gj (x).
j=1
Notice that
j+1
{x ∈ Ωj : f (x) 6= g(x)} ⊆
[
{x ∈ Ωj : f (x) 6= gi (x)}.
i=j−1
This implies that
n
L ({x ∈ Ω : f (x) 6= g(x)}) ≤ 3
∞
X
j=1
n
L ({x ∈ Ωj+2
∞
X
ε
: f (x) 6= gj (x)}) ≤ 3
≤ ε,
6j
j=1
and concludes the proof of Theorem 1.
Corollary 2. Let f : Rn → R be a measurable function, and define Ω = {x ∈ Rn : D − f (x) 6=
∅}. Then for every ε > 0 there exists a function g ∈ C 1 (Rn ) such that
Ln ({x ∈ Ω : f (x) 6= g(x)}) ≤ ε.
Here D − f (x) denotes the Fréchet subdifferential of f at x, that is the set of vectors ζ ∈ Rn
such that
f (x + h) − f (x) − hζ, hi
lim inf
≥ 0.
h→0
|h|
Remark 3. In the above corollary we also have D − f (x) = {∇g(x)} for almost every x ∈ Ω
with f (x) = g(x).
Indeed, almost every point of the set A = {x ∈ Ω : f (x) = g(x)} is a point of density 1 of A,
and for every such point x and every ξ ∈ D − f (x) we have
f (y) − f (x) − hξx , y − xi
g(y) − g(x) − hξx , y − xi
0 ≤ lim inf
= lim inf
,
y→x,y∈A
y→x,y∈A
|y − x|
|y − x|
and
g(y) − g(x) − h∇g(x), y − xi
lim
≥ 0,
y→x,y∈A
|y − x|
hence also
h∇g(x) − ξx , y − xi
≥ 0,
lim inf
y→x,y∈A
|y − x|
which (because x is a point of density 1 of A and h 7→ h∇g(x) − ξx , hi is linear) implies that
−|∇g(x) − ξx | ≥ 0,
that is ∇g(x) = ξx .
For the case k = 2 we have the following.
PROXIMAL SUBDIFFERENTIABLE FUNCTIONS SATISFY A LUSIN PROPERTY OF CLASS C 2
5
Theorem 4. Let Ω ⊂ Rn be a Lebesgue measurable set, and f : Ω → R be a function such that
for almost every x ∈ Ω there exists a vector ξx ∈ Rn such that
f (y) − f (x) − hξx , y − xi
(3)
lim inf
> −∞.
y→x
|y − x|2
Then for every ε > 0 there exists a function g ∈ C 2 (Rn ) such that
Ln ({x ∈ Ω : f (x) 6= g(x)}) ≤ ε.
Proof. As in the proof of Theorem 1 we may assume that Ω is compact and the limiting condition
holds for every x ∈ Ω. Then the assumption implies that
f (y) − f (x)
> −∞
lim inf
y→x
|y − x|
for all x ∈ Ω, so by Theorem 1, given ε > 0, there exists a function g ∈ C 1 (Rn ) such that
ε
Ln ({x ∈ Ω : f (x) 6= g(x)}) ≤ .
4
Observe also that the set A = {x ∈ Ω : f (x) = g(x)} is measurable, so we can find a compact
subset K of A such that Ln (A \ K) ≤ ε/4. Then by replacing Ω with this compact set K, we
may and do assume in what follows that f ∈ C 1 (Rn ) and (6) holds for every x ∈ Ω with ∇f (x)
in place of ξx .
Now let us define for each j ∈ N
Ej := x ∈ Ω : f (y) − h∇f (x), yi ≥ f (x) − h∇f (x), xi − j|y − x|2 for all y ∈ Ω ,
and note that the sets Ej are measurable and increasing to Ω. There exists j0 ∈ N such that
ε
Ln (Ω \ Ej0 ) ≤ .
4
It will be enough for us to prove the following Claim:
|f (y) − f (x) − h∇f (x), y − xi|
< +∞ for Ln − almost every x ∈ Ej0 .
(4)
lim sup
2
|y
−
x|
y→x, y∈Ej0
Assuming for a moment that the Claim is true, and applying Liu-Tai’s result in [18], we obtain
a function h ∈ C 2 (Rn ; R) such that
ε
Ln ({x ∈ Ej0 : f (x) 6= h(x)}) ≤ ,
4
and we easily conclude that
Ln ({x ∈ Ω : f (x) 6= h(x)}) ≤ ε.
In order to prove our Claim (4) we will borrow some ideas from [16]. We define new functions
fe : Rn → R and fˆ : Rn → R by
fe(x) = f (x)n+ j0 |x|2 ,
x ∈ Rn
o
fˆ(x) = sup p(x) : p affine and p ≤ fe on Ω , x ∈ Rn
Using that fe(y) ≥ fe(x) + h∇fe(x), y − xi for all y ∈ Ω it is easy to see that
fe(x) = fˆ(x)
6
D. AZAGRA, J. FERRERA, M. GARCÍA-BRAVO, AND J. GÓMEZ-GIL
for all x ∈ Ej0 . On the other hand, since we are assuming that Ω is compact and f is continuous
on Ω, it is easy to see that fˆ is everywhere finite. Moreover, as sup of affine functions, fˆ is
convex. Therefore fˆ is locally Lipschitz on Ω. Also we are assuming that fe is of class C 1 , hence
so is fe. Since the functions fe and fˆ agree on Ej0 , we then also have that
∇fˆ(x) = ∇fe(x)
for almost every x ∈ Ej0 (see [13, Theorem 3.3(i)] for instance).
Next, by applying Alexandrof’s theorem with the convex function fˆ, we obtain that fˆ is twice
differentiable almost everywhere on Ω. This implies that
(5)
|fe(y) − fe(x) − h∇fe(x), y − xi|
|fˆ(y) − fˆ(x) − h∇fˆ(x), y − xi|
lim sup
=
lim
sup
< +∞
|y − x|2
|y − x|2
y→x, y∈Ej0
y→x, y∈Ej0
for almost every x ∈ Ej0 . However, by the definition of fe(x) = f (x) + j0 |x|2 , we have
|f (y) − f (x) − h∇f (x), y − xi + (j0 (|y|2 + |x|2 − 2hx, yi)|
|f (y) − f (x) − h∇f (x), y − xi|
≤
+ j0 =
|y − x|2
|y − x|2
|fe(y) − fe(x) − h∇fe(x), y − xi|
=
+ j0 ,
|y − x|2
and by combining with (5) we immediately obtain (4).
Corollary 5. Let Ω ⊂ Rn be a Lebesgue measurable set, and f : Ω → R be a function such
that for almost every x ∈ Ω there exists a vector ξx ∈ Rn such that
(6)
lim sup
y→x
f (y) − f (x) − hξx , y − xi
< +∞.
|y − x|2
Then for every ε > 0 there exists a function g ∈ C 2 (Rn ) such that
Ln ({x ∈ Ω : f (x) 6= g(x)}) ≤ ε.
This is of course an immediate consequence of the above theorem applied to −f . According to
Remark 3, we also have that
ξx = ∇g(x)
for almost every x ∈ Ω with f (x) = g(x).
Corollary 6. Let f : Rn → R be a measurable function, and define Ω = {x ∈ Rn : ∂P f (x) 6= ∅}.
Then for every ε > 0 there exists a function g ∈ C 2 (Rn ) such that
Ln ({x ∈ Ω : f (x) 6= g(x)}) ≤ ε.
Here ∂P f (x) denotes the proximal subdifferential of f at x, which is defined as the set of all
ζ ∈ Rd for which there exist σ, η > 0 such that
f (y) ≥ f (x) + hζ, y − xi − σ|y − x|2
for all y ∈ B (x, η). The set ∂P f (x) coincides with {ζ ∈ Rd : ζ = ∇ϕ(x), ϕ ∈ C 2 (Rd ), f − ϕ
attains a minimum at x}, so every function f for which the viscosity subdifferential of second
PROXIMAL SUBDIFFERENTIABLE FUNCTIONS SATISFY A LUSIN PROPERTY OF CLASS C 2
7
order is nonempty at x also has a nonempty proximal subdifferential at x. The set ∂P f (x) can
also be equivalently defined as the set of vectors ζ ∈ Rn such that
lim inf
h→0
f (x + h) − f (x) − hζ, hi
> −∞,
|h|2
so it is clear that the above Corollary is an immediate consequence of Theorem 4. Notice also
that this corollary allows us to recover, with a different proof, the mentioned result for convex
functions established independently by Alberti [2] and Imonkulov [15].
Let us finally present two examples. The first one concerns the following matter: one could
erroneously think that if a function f satisfies (6) then f will automatically satisfy
(7)
lim sup
y→x
f (y) − f (x) − hξx , y − xi
< +∞
|y − x|2
for almost every x ∈ Ω as well, and then one could immediately apply Liu-Tai’s theorem [18]
to conclude the proof of Theorem 4. This is not feasible.
Example 7. Let us consider a Cantor set of positive measure, C ⊂ [0, 1]. More precisely, write
[
C = [0, 1] \ Jn
n
where each Jn is the union of 2n−1 intervals of length
Jn =
n−1
2[
1
.
4n
We denote
Ink ,
k=1
and Ink = (akn , bkn ). Then we define a function f in the following way:
f (x) = 0 for every x ∈ C,
and for every n, k ∈ N, and for every n ∈ N and k = 1, . . . , 2n−1 f : [akn , bkn ] → R will be a non
negative continuous function, such that f : (akn , bkn ) → R is C ∞ ,
1
1
max f (x) = f (akn + (bkn − akn )) = n ,
k
2
2
x∈In
and such that f , as well as all its one-sided derivatives, equals 0 at akn and at bkn .
It is clear that f is continuous. Let us denote
∆x (y) =
f (y) − f (x) − ξx (y − x)
|y − x|2
If x 6∈ C then, taking ξx = f ′ (x), we have limy→x ∆x (y) = 12 f ′′ (x). If x ∈ C, then
f (y) − f (x)
≥ 0.
|y − x|2
Hence for every x there exists ξx such that
lim inf ∆x (y) > −∞.
y→x
8
D. AZAGRA, J. FERRERA, M. GARCÍA-BRAVO, AND J. GÓMEZ-GIL
Let us observe that f also satisfies conditions of the form
lim inf
y→x
f (y) − P (y − x)
> −∞,
|y − x|k
where P is a polynomial of degree k − 1 for every k.
Let C̃ = C \ {akn , bkn }n,k . We claim that
lim sup ∆x (y) = +∞
y→x
for every x ∈ C̃ and every ξx . Let us prove this.
First, we observe that
n
[
[0, 1] \
Jk
k=1
n
consists in 2 intervals of the same length, namely
L [0, 1] \
n
[
Jk = 1 −
k=1
n
X
2k−1
k=1
1
2n+1
+
1
,
22n+1
since
1
1
1
1
= 1 − (1 − n ) = + n+1 .
2
2
2 2
4k
r
k
Hence, if x ∈ C̃, there exist subsequences {amj j }j and {bnjj }j , decreasing and increasing respectively, such that
lim armj j = lim bknjj = x.
j
j
More precisely
|armj j − x| ≤
1
2mj +1
+
1
22mj +1
|bknjj − x| ≤
,
k
k
1
2nj +1
+
1
22nj +1
.
k
Let us consider ξx ≥ 0. We take yj = bnjj − 21 (bnjj − anjj ). We have
∆x (yj ) ≥
1
1
f (yj )
= nj
≥ 2nj
2
|yj − x|
2 |yj − x|2
since |yj − x| ≤ 2n1j .
r
r
r
The case ξx ≤ 0 can be dealt with similarly by considering yj = amj j + 21 (bmj j − amj j ).
Observe that, by Theorem 4, f satisfy the Lusin property of class C 2 . In fact it is not difficult
to find directly a C 2 function which agrees with f up to a set of small measure. Indeed, for a
given ε > 0, we take n0 such that
[
Jn < ε,
L
n>n0
and define g(x) as f (x) if
x∈
n0
[
k=1
Jk
and 0 otherwise. It is clear that g is C and L {x : f (x) 6= g(x)} < ε. In other words, the
function f satisfies the Lusin property of class C ∞ , therefore it is approximately C ∞ almost
everywhere.
∞
PROXIMAL SUBDIFFERENTIABLE FUNCTIONS SATISFY A LUSIN PROPERTY OF CLASS C 2
9
Our second example shows that there are no analogues of Theorem 4 for higher order of differentiability.
Example 8. Let f : R → R be a C 2 function such that f ′′ is not differentiable at any point.
For every x, we have that
f (y) − f (x) − f ′ (x)(y − x) − 12 f ′′ (x)(y − x)
lim
=0
y→x
|y − x|2
If P0 is a positive-definite 2-homogeneous polynomial, we have
lim sup
y→x
f (y) − f (x) − f ′ (x)(y − x) − ( 21 f ′′ (x) + P0 )(y − x)
< 0,
|y − x|2
hence
f (y) − f (x) − f ′ (x)(y − x) − ( 12 f ′′ (x) + P0 )(y − x)
= −∞ < ∞
lim sup
|y − x|k
y→x
for every k > 2. Similarly
f (y) − f (x) − f ′ (x)(y − x) − ( 21 f ′′ (x) − P0 )(y − x)
= ∞ > −∞.
lim inf
y→x
|y − x|k
If a third-order analogue of Theorem 4 were true for this function f , then, according to LiuTai’s characterization of Lusin properties and approximately differentiability of higher order
[18], we would have that f is approximately differentiable of order 3, which would imply by [14,
Theorem 4] that f ′′′ (x) exists for some point x, contradicting the assumption on f .
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ICMAT (CSIC-UAM-UC3-UCM), Departamento de Análisis Matemático, Facultad Ciencias Matemáticas, Universidad Complutense, 28040, Madrid, Spain
E-mail address: [email protected]
IMI, Departamento de Análisis Matemático, Facultad Ciencias Matemáticas, Universidad Complutense, 28040, Madrid, Spain
E-mail address: [email protected]
ICMAT (CSIC-UAM-UC3-UCM), Calle Nicolás Cabrera 13-15. 28049 Madrid, Spain
E-mail address: [email protected]
Departamento de Análisis Matemático, Facultad Ciencias Matemáticas, Universidad Complutense,
28040, Madrid, Spain
E-mail address: [email protected]