A div-curl decomposition for the local
Hardy space
(to appear in the Proceedings of the American Mathematical Society)
Der-Chen Chang, Georgetown University, Department of Mathematics, Washington, DC 20057, USA
,
chang@georgetown.edu∗Galia Dafni, Concordia University, Department of Mathematics & Statistics, Montreal, Quebec, H3G-1M8, CANADA
,
gdafni@mathstat.concordia.ca† Hong Yue, Trine University, Department of Mathematics & Informatics,Angola, IN 46703, USA
,
yueh@trine.edu‡June 11, 2009
Abstract
A decomposition theorem for the local Hardy space of Goldberg, in terms of nonhomogeneous div-curl quantities, is proved via a dual result for the space bmo.
1
Introduction
Given vector fields V = (v1, . . . , vn) in Lp(Rn, Rn), W = (w1, . . . , wn) in
Lp0
(Rn
, Rn) with 1 < p < ∞, 1 p +
1
p0 = 1, the scalar (dot) product V · W =
P viwiwill lie in L1(Rn). Coifman, Lions, Meyer, and Semmes [CLMS] showed
that if the following conditions hold in the sense of distributions:
div V :=X∂vi ∂xi = 0, curl W := ∂wj ∂xi −∂wi ∂xj ij = 0,
∗Partially supported by a Hong Kong RGC competitive earmarked research grant #600607
and a competitive research grant at Georgetown University.
†Partially supported by the Natural Sciences and Engineering Research Council, Canada. ‡Partially supported by the Natural Sciences and Engineering Research Council,
then V · W belongs to the real Hardy space H1(Rn), a proper subspace of L1. Moreover,
kV · WkH1 ≤ CkVkLpkWkLp0. (1.1)
This “div-curl lemma” and other results of a similar nature illustrate the re-cent use of Hardy spaces for applications to nonlinear PDE and in the method of compensated compactness, originally going back to the work of Murat and Tartar.
Recall (see [FS]) that a function f belongs to H1
(Rn) if the maximal function
Mϕ(f ) belongs to L1(Rn), where, for a fixed Schwartz function ϕ, R ϕ = 1, we
define
Mϕ(f )(x) = sup t>0
|f ∗ ϕt(x)|, ϕt(·) = t−nϕ(t−1·).
Here the norm, defined by kf kH1 := kMϕkL1, depends on the choice of ϕ, but
the space does not since different choices of ϕ give equivalent norms. Functions in the Hardy space enjoy both improved integrability and cancellation conditions compared to L1functions. In particular, if f ∈ H1then its integral must vanish.
The local real Hardy space h1
(Rn), defined by Goldberg [Go], is larger than
H1 and allows for more flexibility, since global cancellation conditions are not
necessary. For example, the Schwartz space is contained in h1 but not in H1,
and multiplication by cut-off functions preserves h1but not H1, thus making it
more suitable for working in domains and on manifolds. For membership of a function f in h1(Rn), we use a “local” maximal function mϕ(f ), with
mϕ(f )(x) = sup 0<t<1
|f ∗ ϕt(x)|,
and require mϕ(f ) ∈ L1(Rn). As for H1, kf kh1 := kmϕkL1 defines a norm, and
different choices of ϕ give equivalent norms, so we can choose ϕ with compact support, making clear the local nature of the maximal function.
In [D], sufficient nonhomogeneous conditions on the divergence and curl were found in order for the dot product V · W, where V and W are vector fields as above, to be in h1
(Rn). In particular, as a corollary, the following special case
was proved:
Proposition 1.1 ([D]) Suppose V and W are vector fields on Rn satisfying
V ∈ Lp(Rn)n, W ∈ Lp0(Rn)n, 1 < p < ∞, 1 p+
1 p0 = 1.
If there exists a function f in Lp(Rn) and a matrix-valued function A with components in Lp0(Rn) such that, in the sense of distributions,
div V = f, curl W = A,
then V · W belongs to the local Hardy space h1
(Rn) with
kV · Wkh1≤ C (kVkLpkWkLp0 + kf kLpkWkLp0+ kVkLp
X
i,j
Note that the requirement on the divergence and the components of the curl to be functions in Lp and Lp0, respectively, was also used in the original div-curl lemma of Murat [Mu], and is a natural relaxation of the vanishing divergence and curl conditions. In fact, by the Hodge decomposition, this can be viewed as a combination of the following two theorems:
Theorem 1.2 ([D]) Suppose V and W are vector fields on Rn satisfying
V ∈ Lp(Rn)n, W ∈ Lp0(Rn)n, 1 < p < ∞, 1 p+ 1 p0 = 1 and div V = f ∈ Lp(Rn), curl W = 0
in the sense of distributions. Then V·W belongs to the local Hardy space h1
(Rn)
with
kV · Wkh1(Rn)≤ C (kVkLp(Rn)+ kf kLp(Rn)) kWkLp0(Rn); (1.2)
and
Theorem 1.3 ([D]) Suppose V and W are vector fields on Rn satisfying
V ∈ Lp(Rn)n, W ∈ Lp0(Rn)n, 1 < p < ∞, 1 p+
1 p0 = 1.
If A is a matrix with components in Lp0
(Rn) and
div V = 0, curl W = A
in the sense of distributions, then V · W belongs to the local Hardy space h1
(Rn) with kV · Wkh1(Rn)≤ C kVkLp(Rn) h kWkLp0(Rn)+ X i,j kAijkLp0(Rn) i . (1.3)
It may seem that the conditions on the divergence and the curl in the theo-rems above are too strong, so the question arises as to whether one can charac-terize functions in the Hardy space in terms of such div-curl quantities. Such a characterization, providing a kind of converse to the div-curl lemma, was shown in [CLMS] for H1(Rn):
Theorem 1.4 ([CLMS]) Every function f ∈ H1(Rn) can be written as
f =
∞
X
k=1
λkVk· Wk,
with {λk} ∈ `1and Vk, Wk vector fields with norm bounded by 1 in L2(Rn, Rn),
This decomposition was proved in [CLMS], via functional analysis argu-ments, from the following dual result: for g ∈ BMO(Rn),
kgkBMO≈ sup
V,W
Z
Rn
gV · W, (1.4)
where the supremum is taken over all vector fields V, W in L2
(Rn
, Rn), kVk
L2, kWkL2 ≤
1, satisfying div V = 0, curl W = 0 in the sense of distributions on Rn.
The goal of this paper is to prove an analogue, Theorem 2.2, of (1.4) for functions in bmo(Rn), the dual of the local Hardy space h1(Rn), and conse-quently a decomposition theorem, Theorem 3.1, for h1in terms of the div-curl quantities used in Theorem 1.2 or in Theorem 1.3.
2
The div-curl lemma for local BMO
In [Go] it was shown that the dual of h1(Rn) can be identified with the space bmo(Rn), consisting of locally integrable functions f with
kf kbmo:= sup |I|≤1 1 |I| Z I |f − fI| + sup |I|>1 1 |I| Z I |f | < ∞.
Here the supremum can be taken over balls or cubes with sides parallel to the axes, |I| denotes Lebesgue measure (volume) and fI is the mean of f over I,
i.e. |I|1 R
If . The number 1 in the definition can be replaced by any other finite
nonzero constant. Note that unlike the case of BMO, we do not need to consider this norm modulo constants.
Before stating the main result, let us introduce the following definition in order to simplify notation.
Definition 2.1 Denote by DC1,0p the collection of all functions which can be written in the form V·W, where V, W are vectors fields, V ∈ Lp
(Rn
, Rn), W ∈
Lp0
(Rn
, Rn), kVk
Lp≤ 1, kWkLp0 ≤ 1, satisfying, in the sense of distributions
on Rn,
div V = f ∈ Lp(Rn), kf kLp ≤ 1, and curl W = 0. (2.1)
The collection DC0,1p can be defined analogously by requiring the vector fields to satisfy, instead of (2.1), the conditions
div V = 0, (curl W)ij = Aij∈ Lp
0
(Rn), kAijkLp0 ≤ 1, i, j ∈ {1, . . . , n}. (2.2)
Note that by Theorems 1.2 and 1.3, both DC1,0p and DC0,1p are subsets of h1(Rn) for every 1 < p < ∞.
Theorem 2.2 If g ∈ bmo(Rn), then for 1 < p < ∞, kgkbmo≈ sup f ∈DC1,0p Z Rn gf, (2.3) and kgkbmo≈ sup f ∈DC0,1p Z Rn gf, (2.4)
with constants that depend only on p and the dimension.
Proof: Let g ∈ bmo(Rn) (we are assuming that g is real-valued) and take
f ∈ DC1,0p ∪ DC0,1p . As stated above, by Theorems 1.2 and 1.3, f belongs to h1
(Rn
) with norm bounded by a constant. The duality of bmo(Rn) with h1
(Rn)
then gives
Z
Rn
gf ≤ Ckgkbmo.
It remains to prove the other direction, i.e.
kgkbmo ≤ C0
Z
Rn
gf
where we take the supremum over f ∈ DC1,0p or f ∈ DC0,1p , respectively. We will use the definition of bmo with the supremum taken over balls. Case I: If B is a small ball, say with radius bounded by 1 (although in fact the proof is independent of the radius), one can use the following estimate from the proof of Theorem III.2 in [CLMS]:
1 |B| Z B g(x) − gB 2 dx 1/2 ≤ Cn sup Z g V · W, (2.5)
where the supremum is taken over all vector fields V, W ∈ C0∞( eB) (here and below eB denotes the ball with the same center and twice the radius of B) with kVkLp ≤ 1, kWkLp0 ≤ 1 and div V = 0, curl W = 0. Note that the result in
[CLMS] is stated for a cube Q (instead of a ball B) and for p = p0= 2, but can be modified to the case of a ball and for p 6= 2 as follows.
The key inequality used in the proof in [CLMS],
kg − gBkL2(B)≤ C n X i=1 ∂g ∂xi W−1,2(B) , (2.6)
is valid for any Lipschitz domain (see for example [GR], Section I.2.1, Corollary 2.1). This inequality holds in the homogeneous sense, modulo constants. There-fore, while we denote by W−1,2(B) the dual of the Sobolev space W01,2(B), which is the closure of C0∞(B) under the norm kϕkW1,2(B)= kϕkL2(B)+k∇ϕkL2(B), to
obtain the right-hand-side of (2.6) we test against test functions only bounded in the homogeneous Sobolev norm kukW˙1,2(B)= k∇ukL2(B). For a fixed ball the
homogeneous and nonhomogeneous norms on C0∞(B) functions are equivalent, but here we use only the homogeneous norm in order for the constant to be independent of the radius of the ball.
To estimate ∂g ∂xi
W−1,2(B), i = 1, . . . , n, we fix a j ∈ {1, . . . , n} \ {i} and
define the vector fields V and W, for the case p ≤ 2, as in [CLMS]: given u ∈ C0∞(B) with k∇ukL2≤ 1, let
V = |B|1/2−1/p ∂u ∂xi ej− ∂u ∂xj ei , (2.7)
where ei denotes the ith coordinate vector in Rn, and let
W = γ|B|−1/p0∇ (xj− x0j)η x − x0 R , (2.8)
where x0 and R are the center and radius of B, respectively. Here η is a fixed smooth function supported in B(0, 2) and identically equal to 1 on B(0, 1).
Note that since p ≤ 2, the support and bound on the L2 norm of ∇u imply
kVkLp ≤ 1, while γ can be chosen, depending only on η, n and p, so that
kWkLp0 ≤ 1. This gives us the desired properties for V and W, and moreover,
V · W = γ|B|−1/2∂u ∂xi
. (2.9)
When p ≥ 2, we need to change the definitions of V and W. As above, we take u ∈ C0∞(B) with k∇ukL2 ≤ 1, but now we set
W = |B|1/2−1/p0∇u (2.10) and (again taking j ∈ {1, . . . , n} \ {i})
V = γ0|B|−1/p ∂(ηBxj) ∂xj ei− ∂(ηBxj) ∂xi ej , (2.11) with ηB defined by ηB(x) = η x−x0 R
for η as above. Clearly div V = 0 and curl W = 0, and by the conditions on u, the fact that p ≥ 2, and the choice of γ0, we can make kVkLp ≤ 1 and kWkLp0 ≤ 1. On B, where ηB is identically
equal to 1, we get V = γ0|B|−1/pe
i, so that as before
V · W = γ0|B|−1/2∂u ∂xi
. (2.12)
Integrating against g in (2.9) or (2.12), we can proceed for any p ∈ (1, ∞). Taking the supremum over all such u, we get a bound on
∂g ∂xi W−1,2(B) by a
constant multiple of |B|1/2 times the right-hand-side of (2.5). We then obtain
Case II: Now let us consider a large ball B with radius greater than 1. For this type of ball we need to show a stronger condition, that is, we need to bound the mean of |g| on B. We will do this by showing two cases of the following inequality, corresponding to (2.3) and (2.4):
1 |B| Z B |g(x)|rdx1/r≤ C n,r sup Z g V · W. (2.13)
In both cases the supremum is to be taken over pairs of smooth vector fields V, W supported in ˜B, V ∈ (Lp)n, W ∈ (Lp0)n, with norms bounded by 1, and
in the first case, with r = p0, the vector fields satisfy condition (2.1), while in second case, with r = p, they satisfy (2.2).
Note that a priori we have the finiteness of the left-hand-side of (2.13) by the fact that g ∈ bmo ⊂ BMO ⊂ Lrloc for any r < ∞.
We first prove estimate (2.13) when B = B1is the unit ball B(0, 1). In order
to do this we need to use the full (nonhomogeneous) version of inequality (2.6), namely kgkLr(B1)≤ C ( kgkW−1,r(B1)+ n X i=1 ∂g ∂xi W−1,r(B 1) ) (2.14)
for any r, 1 < r < ∞ (see [Ne], Theorem 1, p. 108, with l = 0). The estimates for
∂g ∂xi W−1,r(B 1)
are analogous to those above (for a fixed ball the norm is equivalent to the homogeneous case). Here the test functions are in W01,r0(B1), so r0 plays the role of the exponent 2 in the argument above,
and again we need to distinguish the two cases. For r = p0, given u ∈ C0∞(B) with k∇ukLp ≤ 1, we use the same vector fields as defined in (2.7) and (2.8),
normalized for the unit ball B1. For r = p the test function u will now have
k∇ukLp0 ≤ 1, so we can define the vector fields as in (2.11) and (2.10).
Now we address the part that is different from the homogeneous case - we have to bound kgkW−1,r(B 1)= sup Z gϕ , (2.15)
where the supremum is taken over ϕ ∈ C0∞(B1), kϕkW1,r0(B1)≤ 1. We need to
be able to write any such function ϕ in terms of div-curl quantities V · W.
Lemma 2.3 If ϕ ∈ C∞
0 (B1) then we can write
ϕ = CV1· W1= CV2· W2
for smooth vector fields Vi, Wi with V1and W2supported in B1, W1 and V2
supported in the double ball fB1 = B(0, 2), W1 ∈ (Lp
0
( fB1))n, V2 ∈ (Lp( fB1))n
(with norms bounded by a constant independent of ϕ), and satisfying
div V2= 0, curl W1= 0.
(i) V1∈ (Lp(B1))n and div V1∈ Lp(B1) with bounds
kV1kLp≤ CkϕkLp, kdiv V1kLp≤ Ck∇ϕkLp;
(ii) W2∈ (Lp
0
(B1))n and, for all i, j, (curl W2)ij ∈ Lp
0
(B1) with bounds
kW2kLp0 ≤ CkϕkLp0, k(curl W2)ijkLp0 ≤ Ck∇ϕkLp0.
Proof: [Proof of Lemma 2.3:] Let ϕ ∈ C0∞(B1). Take
V1= ϕe1= (ϕ, 0, . . . , 0),
and
W1= ∇(ηx1),
where η is supported in fB1= B(0, 2), satisfies kηkL∞ ≤ 1, k∇ηkL∞ ≤ 1, and is
identically equal to 1 on the support of ϕ. Note that this ensures that on B1,
W1= e1, so we get the desired identity
V1· W1= ϕ.
We can bound the norm of W1 by a constant:
kW1kLp0(B1)≤ k∇ηk∞ Z f B1\B1 |x1|p 0 !1/p0 + kηk∞|B1|1/p 0 ≤ Cn,p,
and since W1 is a gradient, we also have curl W1= 0.
Moreover kV1kLp(B 1)≤ kϕkLp(B1) and kdiv V1kLp(B1)= ∂ϕ ∂x1 Lp(B1) ≤ k∇ϕkLp(B1).
For the other case, we can take
V2= ∂(ηx1) ∂x1 e2− ∂(ηx1) ∂x2 e1= −x1 ∂η ∂x2 , x1 ∂η ∂x1 + η, 0, . . . , 0
with η as above. Then
kV2kLp( fB
1). k∇ηkLp( fB1)+ kηkLp( fB1)≤ Cn,p
and div V2= 0. We also set
W2= ϕe2,
so as above, since η is identically 1 on the support of ϕ, we have
In addition,
kW2kLp0(B
1)≤ kϕkLp0(B1)
and
k(curl W2)ijkLp0(B1)≤ k∇ϕkLp0(B1).
Continuation of the proof of Theorem 2.2: We obtain estimate (2.13) for the unit ball, with r = p0 and condition (2.1) satisfied, by using (2.14), applying the lemma with the vector fields V1and W1 to the test functions in (2.15), and
dividing by the appropriate constants. The case r = p and (2.2) corresponds to using the lemma with V2 and W2.
Now let us prove estimate (2.13) for a ball B = B(x0, R), R ≥ 1. On the
left-hand-side, any g ∈ Lr(B) corresponds in a one-to-one and onto fashion to
a ˜g ∈ Lr(B 1) by taking ˜g(x) = g(x0+ Rx), with 1 |B1| Z B1 |˜g(x)|rdx 1/r = 1 |B| Z B |g(y)|rdy1/r.
On the right-hand-side, for any smooth vector fields Vi, Wi corresponding
to the unit ball, as in Lemma 2.3 (i = 1 in the case r = p0, i = 2 in the case r = p), define
V0= Vi(R−1(x − x0)), W0= Wi(R−1(x − x0)).
Then V0, W0are smooth vector fields supported in eB = B(x0, 2R), V0∈ (Lp)n,
and W0∈ (Lp 0 )n with bounds kV0kLp= Z |Vi(R−1(x − x0))|pdx 1/p = Rn/pkVikLp≤ Rn/p, and similarly kW0kLp0 ≤ Rn/p 0 . Moreover, if i = 1 we have div V0∈ Lp,
kdiv V0kLp= Z 1 R(div V1) x − x0 R p dx 1/p = Rn/p−1kdiv V1kLp≤ Rn/p−1, and curl W0= R−1(curl W1)(R−1(x − x0)) = 0,
while if i = 2 we have div V0= 0 and, for all j, k ∈ {1, . . . , n},
k(curl W0)jkkLp= Z 1 R((curl W2)jk) x − x0 R p0 dx 1/p0 ≤ Rn/p0−1.
Letting V = R−n/pV0, W = R−n/p
0
W0and using the fact that R ≥ 1, we
get vector fields satisfying either (2.1) in the first case, or (2.2) in the second case, and Z g V · W = R−n Z ˜ g(R−1(x − x0))Vi(R−1(x − x0)) · Wi(R−1(x − x0))dx = Z ˜ g Vi· Wi.
Taking the supremum over all such Vi, Wi, and using estimate (2.13) for the
unit ball, we get the same inequality for the ball B. This concludes the proof of the theorem.
Remarks:
1. All vector fields constructed in the proof are smooth with compact sup-port, so the suprema on the right-hand-side of (2.3) and (2.4) can be restricted to dot products of such vector fields.
2. As pointed out, the proof of Case I is independent of the radius of the ball and therefore we have generalized the result (1.4) from [CLMS] to p 6= 2. Such a generalization and the resulting decomposition of H1(Rn) in terms of (smooth) div-curl atoms is stated in [BIJZ] (Proposition 2.2) without proof.
3
The decomposition theorem for h
1Since DC1,0p (respectively DC0,1p ) is a bounded symmetric subset of h1(Rn), we can use Lemmas III.1 and III.2 in [CLMS], the duality of bmo and h1[Go], and
Theorem 2.2 to obtain the following decomposition of functions in h1 in terms
of the appropriate “div-curl atoms”:
Theorem 3.1 For a function f in h1
(Rn), 1 < p < ∞, there exists a sequence
{λk} ∈ `1 such that f = ∞ X k=1 λkfk, fk∈ DC1,0p for all k ≥ 1.
Such a decomposition also holds with fk ∈ DC p
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