Mathematische Annalen
Math. Ann. 324, 341–358 (2002) DOI: 10.1007/s00208-002-0341-5
H1 -estimates of Jacobians by subdeterminants Tadeusz Iwaniec · Jani Onninen Received: 20 November 2000 / Revised version: 17 December 2001 / Published online: 5 September 2002 – © Springer-Verlag 2002 1,n−1 (Ω, Rn ), n ≥ 2. We Abstract. Let f : Ω → Rn be a mapping in the Sobolev space Wloc n assume that the cofactors of the differential matrix Df (x) belong to L n−1 (Ω). Then, among other things, we prove that the Jacobian determinant detDf lies in the Hardy space H1 (Ω).
Mathematics Subject Classification (2000): 42B25, 26B10
1. Introduction Let Ω be a nonempty open subset of Rn , n ≥ 2. We study mappings f = 1,n−1 (f 1 , ..., f n ) : Ω → Rn in the Sobolev space Wloc (Ω, Rn ) whose differential matrix Df = [∂f i /∂xj ] satisfies n D f ∈ L n−1 (Ω) (1) where D f denotes the cofactor matrix of Df . The aim of this paper is to answer a question of M¨uller, Qi and Yan [16] by proving the following theorem. Theorem 1.1 The Jacobian determinant J (x, f ) = det Df (x) belongs to the Hardy space H1 (Ω), and we have the uniform estimate n D f (x) n−1 dx (2) ||det Df ||H1 (Ω) ≤ C(n) Ω
Here and in the sequel we shall use a maximal characterization of the space H1 (Ω) over a domain Ω ⊂ Rn . In Example 7.1 we show that the H1 -regularity of Jacobians may fail if one lowers the degree of integrability of the cofactor T. Iwaniec Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA (e-mail:
[email protected]) J. Onninen Department of Mathematics, University of Jyv¨askyl¨a , P.O. Box 35, Fin-40351 Jyv¨askyl¨a, Finland (e-mail:
[email protected].fi) Iwaniec was supported by NSF grant DMS-0070807. This research was done while Onninen was visiting Mathematics Department at Syracuse University. He wishes to thank SU for the support and hospitality.
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matrix. On account of the well known fact [7] that BMO(Rn ) is the dual space to H1 (Rn ), we obtain Corollary 1.2 Under the assumptions of Theorem 1.1, with Ω = Rn , we have n D f (x) n−1 dx ϕ(x)J (x, f ) dx ≤ C(n)||ϕ||BMO(Rn ) (3) Rn
Rn
where ϕ ∈ BMO(R ). n
This integral in the left hand side is understood by means of H1 − BMO duality, see [19, p.142-144]. The reader will notice that the expression in the left hand side of (2) is homogeneous of degree 1 with respect to each of the coordinate functions f 1 , f 2 , . . . , f n , whereas the right hand side is lacking this type of homegeneity. It is, therefore, natural to reformulate Estimate (2) by using the (n − 1)-forms df 1 ∧ ... ∧ df k−1 ∧ df k+1 ∧ ... ∧ df n n ∂ f 1 , ..., f k−1 , f k+1 , ..., f n = dx1 ∧ ... ∧ dxi−1 ∧ dxi+1 ∧ ... ∧ dxn ∂ , ..., x , x , ..., x ) (x 1 i−1 i+1 n i=1 where the coefficients above are made up of the cofactors of Df . Elementary analysis of homogeneity now leads us to the following strenthening of the estimate in Theorem 1.1 det Df H1 (Ω) ≤ Cp (n)
n 1 df ∧ ... ∧ df k−1 ∧ df k+1 ∧ ... ∧ df n
Lpk (Ω)
k=1
(4) where p = (p1 , ..., pn ) is an arbitrary n-tuple of exponents p1 , ..., pn > 1, such that 1 1 + ... + =n−1 p1 pn In a way the H1 -regularity of Jacobians was first observed by Wente [21], though a systematic study begins with the work by Coifman, Lions, Meyer and Semmes [5] and [6]. They proved that the Jacobian determinant, denoted here by J (x, f ) = det Df (x), of a mapping f : Rn → Rn in the Sobolev space W 1,n (Rn , Rn ) lies in the Hardy space H1 (Rn ); also see [15] for the case of nonnegative Jacobians. As a matter of fact, the Jacobian operator J : W 1,n (Rn , Rn ) → H1 (Rn ) is continuous. We actually have a uniform bound ||det Df − det Dg||H1 (Rn ) ≤ C(n)||Df − Dg||n (||Df ||n + ||Dg||n )n−1 (5) Since then there has been dramatic development made in the H1 -theory of Jacobians and related nonlinear differential forms. Most notable subsequent publications are [9], [4], [13] and [16]. In [16] the authors considered orientation
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1,n−1 preserving mappings in Wloc (Ω, Rn ) which satisfy Condition (1). The term orientation preserving pertains to the mappings whose Jacobian determinant is nonnegative. Among numerous results in [16] it is shown that the Jacobian belongs to the Zygmund class L log L(Ω ), on every relatively compact subdomain Ω ⊂ Ω. In particular, it belongs to H1 (Ω ). Let us point out that the condition (1) was motivated by a study of the existence problems in nonlinear elasticity [1] and [20]. On the analogy of [6] M¨uller, Qi and Yan raised the question as to whether the Jacobian remains in the Hardy space H1 (Ω ) if one allows it to change sign. Theorem 1.1 answers this question in the affirmative. Of course, as a consequence of a result by Stein [18], we recover that for orientation preserving mappings det Df ∈ L log L(Ω ). The main ingredient to our arguments is the use of spherical maximal inequality, as shown in [17] and [2]. This seems to be the first time that spherical maximal functions have been successfully employed in the study of Jacobians. Combining the uniform estimate (2) with the fact that H1 (Rn ) is the dual of VMO(Rn ) we conclude the result concerning convergence of Jacobians in the weak star topology of H1 (Rn ).
Theorem 1.3 Suppose fj f weakly in W 1,n−1 (Rn , Rn ) and
Rn
n D fj (x) n−1 dx ≤ K
for i = 1, 2, 3, ... Then for every ϕ ∈ VMO(Rn ) we have ϕ(x)J (x, fj ) dx = ϕ(x)J (x, f ) dx lim j →∞
Rn
Rn
(6)
(7)
(8)
Here the functions need not be L1 -integrable, so the meaning of the integrals is understood in the sense of H1 − BMO duality. Example 7.2 will reveal that Jacobians need not converge in the usual weak topology of H1 (Rn ). Precisely, (8) fails for ϕ(x) = log |x|. n In the spirit of [13] we generalize Theorem 1.1 by assuming that |D f | n−1 belongs to an Orlicz space LP (Ω), with some convex function P . In this case, as might to expected, the Jacobians lay in the Hardy-Orlicz space HP (Ω), see Theorem 6.2. Here are some examples: P1 (t) = t logα (1 + t), α ≥ 0
(9)
and P2 (t) = t exp
log(1 + t) log log(e + t)
(10)
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We recall from [13] an analogue of Stein’s result which tells us that a nonnegative P∗ function in HP (Ω) belongs to Lloc (Ω), where
∗
P (t) = t 0
t
P (s) ds 1 + s2
Thus, for nonnegative Jacobians we gain some degree of integrability. For example, P1∗ (t) ≈ P1 (t) log(1 + t) and P2∗ (t) ≈ P2 (t) log log(e + t) Note that we gain nothing if P (t) = t p for some p > 1. Finally, the interested reader may wish to consult the recent paper [8] for further develelopments concerning L1 -integrability of Jacobians of orientation preserving mappings. 2. Maximal operators Let Ω be an open subset of Rn . The Hardy-Littlewood maximal operator is defined on L1loc (Ω) by the rule Mh(x) = MΩ h(x) = sup − |h(y)| dy : 0 < t < dist(x, ∂Ω) (11) B(x,t)
1 where, as usual, −B |h| = |B| B |h| = |h|B stands for the mean value of |h| over the set B. In the above definition, B = B(x, t) is the ball centered at x ∈ Ω and radius t. Most often the dependence of M on the domain Ω will not be emphasized. We record the following local variant of the well know maximal inequality ||Mh||Lp (Ω) ≤ Cp (n)||h||Lp (Ω)
(12)
C(n) . Another important maximal operator for 1 < p ≤ ∞, where Cp (n) ≤ pp−1 was introduced to harmonic analysis by Stein [17]. It involves spherical averages: |h(y)| dy : 0 < t < dist(x, ∂Ω) (13) (Sh) (x) = (SΩ h) (x) = sup − S(x,t)
Here and also in what follows we use the notation S(x, r) = ∂B(x, r). Notice that the integral average of |h| is taken with respect to the (n−1)-dimensional surface area. As shown by Bourgain [2] for n = 2 and Stein [17] in higher dimensions,
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the spherical maximal operator is bounded in Lp -spaces for all p > n for p = n−1 . That is ||Sh||Lp (Ω) ≤ Cp (n)||h||Lp (Ω) ,
p>
n , n−1
n n−1
but not
(14)
We shall now introduce one parameter family {Mθ }θ≥1 of maximal operators Mθ h(x) = sup
n tn
t
0
θ1
θ
n−1 r −
|h(y)|dy
dr
:
(15)
S(x,r)
0 < t < dist(x, ∂Ω)
The Hardy-Littlewood operator is none other than M1 , whereas the spherical operator arises by letting θ go to infinity. In consequence of the maximal inequalities at (12) and (14) we have the following result. Theorem 2.1 The sublinear operator Mθ : Lp (Ω) → Lp (Ω) is bounded for n all p > n−1+ 1 θ
Proof. The case θ = 1 reduces to the Hardy-Littlewood maximal inequality, so we may assume that θ > 1. The condition on p can be rewritten as 1−
n−1 θ −1 p
Now we choose 0 < α < 1, to satisfy 1−
n−1 θ −1 p <α
(16)
By H¨older’s inequality we estimate the spherical averages − S(x,r)
θ |h|
≤ −
αθ
|h| θ −1
S(x,r)
≤ S |h|
αθ θ −1
θ−1 −
θ−1
|h|θ−αθ
S(x,r)
1 nωn r n−1
|h|θ−αθ S(x,r)
Here and in the sequel all maximal functions will be evaluated at x. Substituting this inequality into formula (15) yields θ−1 αθ [Mθ h]θ ≤ S |h| θ −1 M |h|θ−αθ
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Next, we use H¨older’s inequality 1− θ1 1 αθ θ−αθ θ |h| ||Mθ h||p ≤ S |h| θ −1 M p α
1 1 αθ 1− θ θ −1 = S |h| pθ −p M |h|θ−αθ θ p
p 1−α
θ −αθ
αθ
p n Notice that (16) yields pθ−p > n−1 and θ−αθ > 1. This makes it legitimate to αθ apply maximal inequalities at (12) and (14).
αθ 1− 1 1 θ ||Mθ h||p ≤ Cp (n, θ ) |h| θ −1 pθ −p |h|θ−αθ θ p
θ −αθ
αθ
Cp (n, θ ) hpα
= = Cp (n, θ ) hp
hp1−α
as desired. Remark 2.2 Notice that bounded in L
n n−1
n n−1
is always larger than
n n−1+ θ1
, so Mθ is always
(Ω).
3. The maximal operator on distributions We shall rely on one particular approximation of the identity. That is, we fix a radially symmetric function Φ ∈ C0∞ (Rn ) supported in the unit ball and having integral 1. For example exp |x|21−1 if |x| < 1 Φ(x) = C(n) 0 if |x| ≥ 1
where the constant C(n) is chosen so that Φ(x) dx = 1. For each t > 0, we consider one parameter approximation to the Dirac mass Φt (x) = t −n Φ xt . Given h ∈ L1loc (Ω), we recall the mollifiers Φt (x − y)h(y) dy (h ∗ Φt ) (x) = Ω
whenever 0 < t < dist(x, ∂Ω). We mimic this convolution formula to extend the mollification procedure to Schwartz distributions h ∈ D (Ω) as follow (h ∗ Φt ) (x) = h[Φt (x − ·)] where we notice that the function y → Φt (x − y) belongs to C0∞ (Ω) for 0 < t < dist(x, ∂Ω).
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Then the associated maximal function of h can be defined as Mh(x) = MΩ h(x) = sup |h ∗ Φt (x)| : 0 < t < dist(x, ∂Ω) This maximal function works well in connection with H1 -spaces because it takes the possible cancellation into account. Also we like to mention that the estimate Mh(x) ≤ C(n)Mh(x) follows easily from the inequality Φt (x − y)h(y) dy ≤ C(n) |h(y)| dy t n B(x,t) Ω The Hardy space H1 (Ω) consists of distributions h ∈ D (Ω) such that Mh(x) dx < ∞ ||h||H1 (Ω) = Ω
Note that H (Ω) ⊂ L (Ω) and || · ||L1 (Ω) ≤ || · ||H1 (Ω) . For Ω = Rn our definition coincides with the usual definition of the Hardy space H1 (Rn ), see [19]. Various bounds for Mh can be obtain from the following elementary calculation. 1
1
Lemma 3.1 Let B(x, r) denote the ball centered at x ∈ Ω and radius 0 < r < dist(x, ∂Ω). Then C(n) t |h ∗ Φt (x)| ≤ n+1 h dr (17) t 0 B(x,r) Proof. Denote by S(x, r) = ∂B(x, r). By Fubini’s Theorem we obtain Φt (x − y)h(y) dy (h ∗ Φt ) (x) = B(x,t) t = Φt (x − y)h(y) dy dr 0 S(x,r) t Φt (r) h(y) dy dr = 0 S(x,r) t d Φt (r) h(y) dy dr = dr B(x,r) 0 t d =− Φt (r) h(y) dy dr dr 0 B(x,r) t C(n) h dr ≤ n+1 t 0 B(x,r) as desired.
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4. Isoperimetric inequality 1,n−1 (Ω, Rn ) satisfy Condition (1). Our main device to estimating the Let f ∈ Wloc maximal function MJ by the subdeterminants of Df is the following isoperimetric type inequality n n−1 J (y, f ) dy ≤ C(n) (18) D f (y) dy B(x,r)
S(x,r)
for almost every 0 < r < dist(x, ∂Ω). We refer to [16] for the proof. Now Lemma 3.1 gives n n−1 C(n) t |(J ∗ Φt ) (x)| ≤ n+1 dr Df t 0 S(x,r) n n−1 C(n) t n r dr Df − ≤ n+1 t 0 S(x,r) n n−1 C(n) t n−1 D f − r dr ≤ n t 0 S(x,r) n n−1 n |D f | (x) ≤ C(n) M n−1 Taking supremum with respect to 0 < t < dist(x, ∂Ω), we conclude with the pointwise inequality n (19) MJ (x) ≤ C(n) Mθ |D f | n−1 (x) n . for almost every x ∈ Ω, where θ = n−1 Let us strenghten inequality (18) slightly as follows. 1,n−1 Lemma 4.1 For each f ∈ Wloc (Ω, Rn ) satisfying (1) we actually have − J (y, f ) dy (20) B(x,r)
≤ C(n)
n k=1
1 1 n−1 df ∧ ... ∧ df k−1 ∧ df k+1 ∧ ... ∧ df n
S(x,r)
for almost every 0 < r < dist(x, ∂Ω). Proof. This result is a consequence of Inequality (18) by the following analysis of the terms in both sides of (18). First we observe point-wise inequality n 1 df ∧ ... ∧ df k−1 ∧ df k+1 ∧ ... ∧ df n D f ≤ k=1
(21)
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Fubini’s Theorem tells us that for almost every r the integrals in the right hand side of (20) are finite. In what follows we consider only such radii. Fix any positive integer l. For k = 1, 2, ..., n we define the positive numbers
1 + −S(x,r) df 1 ∧ ... ∧ df k−1 ∧ df k+1 ∧ ... ∧ df n l λk = 1 n 1
df 1 ∧ ... ∧ df j −1 ∧ df j +1 ∧ ... ∧ df n n +− j =1
l
S(x,r)
Clearly λ1 · · · λn = 1 and we may apply (18) to the mappings (λ1 f 1 , . . . , λn f n ) in place of (f 1 , . . . , f n ), to obtain − J (y, f ) dy B(x,r) n n n−1 λ−1 df 1 ∧ ... ∧ df k ∧ ... ∧ df n ≤ C(n) − k
S(x,r) k=1
n
n n−1 1 1 + − ≤ C(n) λ−1 df k ∧ ... ∧ df n df ∧ ... ∧ k l S(x,r) k=1 1 n n−1 1 1 n j + − ≤ C(n) · n df ∧ ... ∧ df ∧ ... ∧ df l S(x,r) j =1
As usual, the symbol under is to be omitted. We should remark here that the radii of the balls for which these inequalities hold may depend on the mappings (λ1 f 1 , ..., λn f n ). Since we are dealing with only countable number of such mappings we may assume that inequalities hold for almost every 0 < r < dist(x, ∂Ω), and with every l = 1, 2, ... Now the inequality (20) is established by letting l go to infinity. With this version of isoperimetric inequality we can obtain a better estimate of MJ (x). Indeed, using Inequality (20) instead of (18), we obtain 1 n−1 n C(n) t n 1 n k |(J ∗ Φt ) (x)| ≤ n+1 − |df ∧ ... ∧ df ∧ ... ∧ df | r dr t 0 S(x,r) k=1 1 n−1 n C(n) t n−1 1 n k ≤ n r dr − |df ∧ ... ∧ df ∧ ... ∧ df | t 0 S(x,r) k=1 Then, by H¨older’s inequality |(J ∗ Φt ) (x)| ≤
n C(n) k=1
tn
t
r n−1
0
1 pk p (n−1) k 1 n k − |df ∧ ... ∧ df ∧ ... ∧ df | dr
S(x,r)
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whenever the positive numbers p1 , ..., pn satisfy p11 + ... + supremum with respect to 0 < r < dist(x, ∂Ω) we have
= n − 1. Taking
1 pn
n 1 n−1 MJ (x) ≤ C(n) Mpk (df 1 ∧ ... ∧ df k−1 ∧ df k−1 ∧ ... ∧ df n ) k=1
(22) where
1 p1
+ ... +
1 pn
= n − 1.
5. Proof of Theorems 1.1 and 1.3 We estimate the L1 -norm of MJ by using inequality (19) n MJ L1 (Ω) ≤ C(n) Mθ |D f | n−1n L n−1 (Ω) n n n−1 n−1 D f ≤ C(n) D f n = C(n) L n−1 (Ω)
Ω n
Here, of course, we have used boundednes of the operator Mθ in L n−1 (Ω). This completes the proof of Theorem 1.1. We now proceed to the proof of inequality (4). We integrate both side of (22) and use H¨older’s inequality to obtain n−1 MJ (x) dx ≤
Ω n
pk p1k
Mpk (df ∧ ... ∧ df k ∧ ... ∧ df n )
C(n)
1
Ω
k=1
pk p1k 1 n k ≤ Cp (n) df ∧ ... ∧ df ∧ ... ∧ df Ω
This last inequality follows from the fact that the sublinear operators Mpk are bouded from Lpk (Ω) to Lpk (Ω), provided pk > 1 for k = 1, ..., n, see Theorem 2.1. The proof of inequality (4) is complete. Next we shall prove Theorem 1.3. First observe that Inequality (7) remains valid for the limit mapping, that is: n D f (x) n−1 dx ≤ K (23) Rn
This follows from (6), (7) and the well know fact that D fj → D f in D (Ω), see for example [11, Chapter 8.1]. In particular, by Theorem 1.1, we have det Dfj 1 + det Df H1 ≤ C(n) K (24) H
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for all j = 1, 2, ... Next we recall that the dual to H1 (Rn ) is BMO(Rn ) ⊃ VMO(Rn ). Given ϕ ∈ VMO(Rn ) it is customary to write ϕ(x)h(x) dx ϕ[h] = Rn
Here the integral is nothing but the action of ϕ on h. This action coincides with the integral above if ϕ ∈ C0∞ (Ω). For any > 0 we can find a test function ϕ ∈ C0∞ (Ω) such that ||ϕ − ϕ ||BMO ≤ , as C0∞ (Ω) is dense in VMO(Rn ). To simplify notation it will be convenient to write Jk = J (x, fk ) and J = J (x, f ). Now ϕ[Jk ] − ϕ[J ] = ϕ [Jk ] − ϕ [J ] + (ϕ − ϕ )[Jk ] − (ϕ − ϕ )[J ]
(25)
It follows from (3), (24) and lim ϕ [Jk ] = ϕ [J ] that k→∞
lim sup |ϕ[Jk ] − ϕ[J ]| ≤ lim sup ϕ − ϕ BMO (Jk H1 + J H1 )
k→∞
k→∞
≤ C(n) K As was arbitrary we conclude that lim ϕ[Jk ] = ϕ[J ]
k→∞
completing the proof of Theorem 1.3. 6. Estimates in Hardy-Orlicz norms Let P : [0, ∞] → [0, ∞] be a Young (that is, convex) function. The Orlicz space, by LP (Ω), consists of all measurable funtions u on Ω such
denoted −1 that Ω P (k |u|) < ∞, for some k = k(f ) > 0. We shall make use of the Luxemburg norm P (k −1 |u|) ≤ 1 ||u||P = ||u||LP (Ω) = inf k > 0 : Ω
so LP (Ω) becomes a Banach space. Definition 6.1 The Hardy-Orlicz space HP (Ω) consists of Schwartz distributions u ∈ D (Ω) such that ||u||HP (Ω) = M(u)LP (Ω) < ∞ For more information about Hardy-Orlicz spaces we refer to [14] and [3], see also [13].
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Theorem 6.2 Let P : [0, ∞] → [0, ∞] be a Young function such that t −N P (t) decreases to zero for some N > 1. nConsider a mapping f : Ω → Rn of Sobolev 1,n−1 class Wloc (Ω, Rn ), with |D f | n−1 ∈ LP (Ω). Then its Jacobian belongs to the Hardy-Orlicz space HP (Ω), and we have n J HP (Ω) ≤ Cp (n) |D f | n−1 P L (Ω)
Proof. The proof is much the same as for Theorem 1.1. Here are some details. We estimate the P -norm of MJ by using (19) n MJ P ≤ C(n) Mθ |D f | n−1 P n n−1 = C(n) Mθ |D f |
n
where Q(t) = P t n−1
Q
and θ =
n . n−1
Next notice that the operator Mθ :
Lp (Ω) → Lp (Ω) is bounded for all p > n − n−1
n2 , n2 −1
by Theorem 2.1. Also observe
Q(t) is increasing, whereas t → t − n−1 Q(t) is that the function t → t decreasing. Then by an interpolation argument, see for instance [12], we find that the operator Mθ : LQ (Ω) → LQ (Ω) is also bounded. Therefore n n MJ P ≤ Cp (n) D f Qn−1 = Cp (n) |D f | n−1 nN
P
as desired. 7. Examples This section is dedicated to showing that both Theorem 1.1 and Theorem 1.3 are sharp. The first example demonstrates that one cannot lower the degree of integrability of the cofactor matrix D f and achieve the same conclusion as in Theorem 1.1. Example 7.1 Let Ψ : [0, ∞) → [0, ∞) be any continuously increasing function such that Ψ (t) =0 t→∞ t lim
(26)
Then there exists an orientation preserving mapping F : Ω → Rn in the Sobolev space W 1,n−1 (Ω, Rn ) such that n n−1 Ψ DF Ψ |DF |n < ∞ (27) ≤ Ω
Ω
and its Jacobian determinant fails to belong to H1 (Ω).
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Construction. Let Ω be any nonempty bounded domain in Rn . The mapping we are going to construct will have nonnegative Jacobian, J (x, F ) = detDF (x) ≥ 0, and J (x, F ) log(e + J (x, F )) dx = ∞ (28) Ω
for some relatively compact Ω ⊂ Ω. This will suffice to conclude that det DF ∈ / H1 (Ω), by the well known result of Stein [18]. Given a ball B = {x ∈ Ω : |x − a| < r} and 0 < α < 1, we consider the radial stretching r α (x − a) |x − a|α
(29)
r nα |x − a|nα
(30)
(1 − α)r nα |x − a|nα
(31)
f (x) = a + An elementary computation shows that |Df (x)|n =
J (x, f ) =
where we have used the operator norm of the differential matrix Df (x). Using polar coordinates we compute the integrals: |B| ∞ Ψ (t) Ψ |Df |n = dt α 1 t 1+ α1 B
(32)
∞ 1 1 1 J (x, f ) log(e + J (x, f )) dx = (1 − α) α |B| log(e + t) t − α dt α B 1−α ∞ 1 1 1 ≥ (1 − α) α |B| (log t) t − α dt α 1 α|B| (33) = 1 (1 − α)2− α
Next we notice by using Condition (26) that ∞ Ψ (t) lim (1 − α) 1 dt = 0 α→1 1 t 1+ α
(34)
This allows us to define a sequence {αk }k=1,2,... with 21 ≤ αk < 1, converging to 1, and such that ∞ Ψ (t) 1 (35) (1 − αk ) 1 dt ≤ 1+ k 1 t αk
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for k = 1, 2, .... We shall now consider balls Bk ⊂ Rn whose volumes are defined by the equation 1 − αk |Bk | = k The centers of these balls will be determined later. First notice that ∞ ∞ ∞ ∞ Ψ (t) Ψ (t) |Bk | dt ≤ |B | dt k 3 1+ 1 t 1 1 t αk k=1 k=1 ≤
∞ k=1
π2 |Bk | = k(1 − αk ) 6
(36)
At this point we shall appeal to a geometric fact that balls of finite total volume can be packed in a cube Ω ⊂ Rn . Precisely, we can arrange the centers of Bk = Bk (ak , rk ) to obtain a disjoint family of ball contained in Ω. One can actually obtain a uniform bound for the volume of the cube Ω, ∞ |Ω| ≤ C(n) |Bk | k=1 Let us denote by Ω = ∪∞ k=1 Bk . Of course we may assume that Ω is relatively compact in Ω. This packing problem is generally known and, therefore, we shall only sketch the proof. But before, let us finish the construction. With this arrangement of the centers of the balls we define the mapping F : Ω → Rn by the rule α rk k (x−ak ) a + for x ∈ Bk k |x−ak |αk F (x) = x for x ∈ Ω \ Ω On account of formula (32) we find that ∞ |Bk | ∞ Ψ (t) n Ψ |DF | = <∞ αk 1 t 1+ α1k Ω k=1 by (36). We also have the trivial estimate Ψ |DF |n = Ψ (1) ≤ Ψ (1)|Ω| Ω\Ω
Ω\Ω
which gives (27). To verify (28) we use inequality (33), ∞ αk |Bk | J (x, F ) log(e + J (x, F )) dx ≥ 2− α1 Ω k k=1 (1 − αk ) ∞ −1 αk (1 − αk ) αk 1
=
k
k=1
≥
1 2e
∞ k=1
1 =∞ k
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as desired. Regarding the packing problem, we first consider a cube Ω of side 1. Any cube Ω ⊂ Rn whose volume equals 2−nk , for some k = 0, 1, 2, ..., will be called a dyadic cube. We observe that any collection F = {Qi }i=1,2,... of dyadic cubes whose total volume does not exceed 1 can be packed into Ω. Here is an explanation how to make such packing. For this, we call Q ∈ F a k-cube if |Q| = 2−nk , k = 0, 1, 2, ... Clearly, if F contains a 0-cube then this is the only members of F, which can be packed into Ω. If not, then we divide Ω into 2n equal boxes. Note that we have enough number of such boxes in order to pack all 1-cubes in F. Next, we divide each of the empty boxes into 2n new boxes, each of volume 2−2n . As before, the number of new boxes is sufficient to accomodate all of the 2-cubes in F. Continuing this process, infinitely many times if necessary, we finally pack all cubes of F into Ω. Now suppose that we are given balls Bk , k = 1, 2, ... whose total volume satisfies ∞ n 4n |Bk | ≤ ωn−1 k=1 Here, and in the sequel, ωn−1 denotes the surface area of the unit sphere in Rn . It is not difficult to see that every such ball Bk is contained in a dyadic cube Qk such that ωn−1 |Qk | ≤ |Bk | n 4n Hence the total volume of these dyadic cubes does not exceed 1. Consequently, we can pack all the ball Bk , k = 1, 2, ... into Ω. The general case of the packing problem follows by rescaling. In much the same way as in Example 7.1 one can show that under the condition (1) we cannot deduce HP -regularity of the Jacobian for any Young function P = P (t) such that lim t −1 P (t) = ∞. t→∞ Our second example deals with Theorem 1.3. It demonstrates that the space VMO(Rn ) of test functions cannot be replaced by BMO(Rn ). In other words, the sequence of Jacobians in our example will not converge weakly in H1 (Rn ). Example 7.2 There is a sequence {fk } bounded in W 1,n (Rn , Rn ), converging uniformly to zero, and such that 1 1 J (x, fk ) dx ≥ (37) lim inf log k→∞ |x| n Rn This example is of course stronger than we need. Uniform convergence together with boundedness in W 1,n (Rn , Rn ) implies that fk 0 weakly in W 1,n−1 (Rn , Rn ). Moreover, we have uniform bound n n−1 |Dfk |n < ∞ ≤ sup D fk sup k≥1
Rn
k≥1
Rn
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Note that the function log |x| lies in BMO(Rn ) but not in VMO(Rn ). Construction. Given any positive integer k we consider a radial stretching in the ball B = {x ∈ Rn : |x| ≤ 1e }. gk (x) =
Ck x 1
1
|x| |log |x|| n + k
(38)
,
where Ck is a constant determined by the equation ! n Ck = n k ωn−1
(39)
An elementary computation shows that |Dgk (x)| = and
Ck 1
1
J (x, gk ) =
(40)
1
|x| |log |x|| n + k
+ k1 Ckn n n |x|n |log |x||2+ k
(41)
Using polar coordinates we compute |Dgk (x)|n dx = 1
(42)
B
Hence, by (40) and (41) we also have 1 1 1 1 |Dgk (x)|n dx ≥ J (x, gk ) = + log |x| n k n B B
(43)
We shall now define the desired sequence {fk } of mappings fk : Rn → Rn by extending gk beyond the ball B as follows if 0 < |x| ≤ 1e gk (x) fk (x) = Ck xη(|x|) if 1e < |x| ≤ 1 0 otherwise Here η = η(t) is an arbitrary Lipschitz function defined on the interval [ 1e , 1], such that η 1e = 1, η(1) = 0 and 0 ≤ η(t) ≤ 1. Notice that lim (44) (log |x|) J (x, fk ) dx = A1 lim Ckn = 0 k→∞
|x|≥ 1e
k→∞
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where the constant A1 depends only on the choice of the function η. Combinnig this fact with inequalty (43) we conclude that 1 1 J (x, fk ) dx ≥ lim inf log k→∞ |x| n Rn proving the claim at Inequality (37). Similarly, we find |Dfk (x)|n dx = |Dgk (x)|n dx + A2 Ckn = 1 + A2 Ckn Rn
B
with A2 depending only on η. In particular, the sequence {fk } is bounded in W 1,n (Rn , Rn ). The last thing to show is that |fk (x)| ≤ Ck for all x ∈ Rn and k = 1, 2, ... Thus fk → 0 uniformly in Rn . References 1. Ball, J.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1978), 337–403 2. Bourgain, J.: Estimations de certaines fonctions maximales. (French) [Estimates of some maximal operators] C. R. Acad. Sci. Paris Sr. I Math. 301 (10) (1985), 499–502 3. Bonami, A., Madan, S.: Balayage of Carleson measures and Hankel operators on generalized Hardy spaces. Math. Nachr. 153 (1991), 237–245 4. Coifman, R., Grafakos, L.: Hardy space estimates for multilinear operators. I. (English. English summary) Rev. Mat. Iberoamericana 8 (1992), 45–67 5. Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compacit par compensation et espaces de Hardy. (French) [Compensated compactness and Hardy spaces] C. R. Acad. Sci. Paris Sr. I Math. 309 (18) (1989), 945–949 6. Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72 (9) (1993), 247–286 7. Fefferman, C., Stein, E. M.: H p spaces of several variables. Acta Math. 129 (1972), 137–193 8. Giannetti, F., Iwaniec, T., Onninen, J., Verde, A.: Estimates of Jacobians by Subdeterminants. J. Geom. Anal. 12 (2) (2002), 223–254 9. Grafakos, L.: Hardy space estimates for multilinear operators. II. Rev. Mat. Iberoamericana 8 (1992), 69–92 10. Greco, L., Iwaniec, T., Moscariello, G.: Limits of the improved integrability of the volume forms. Indiana Univ. Math. J. 44 (1995), 305–339 11. Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs, Clarendon Press 2001 12. Iwaniec, T., Sbordone, C.: Quasiharmonic Fields. To appear in Ann. Inst. Poincar`e Analyse Non Lineaire 18 (5) (2001), 519–572 13. Iwaniec, T., Verde,A.:A study of Jacobians in Hardy-Orlicz spaces. Proc. Roy. Soc. Edinburgh Sect.A 129 (3) (1999), 539–570 14. Janson, S.: Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J. 47 (1980), 959–982
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15. M¨uller, S.: Higher integrability of determinants and weak convergence in L1 . J. Reine Angew. Math. 412 (1990), 20–34 16. M¨uller, S., Qi, T.,Yan, B. S.: On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincar´e Anal. Non Linaire 11 (2) (1994), 217–243 17. Stein, E. M.: Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. U.S.A. 73 (7) (1976), 2174–2175 18. Stein, E. M.: Note on the class L log L. Studia Math. 32 (1969), 305–310 19. Stein, E. M.: . Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43 20. Sver`ak, V.: Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988), 105–127 21. Wente, H.: The Dirichlet problem with a volume constraint. Manuscripta Math. 11 (1974), 141–157
