arXiv:1703.00231v1 [math.AP] 1 Mar 2017
FRACTIONAL DIV-CURL QUANTITIES AND APPLICATIONS TO NONLOCAL GEOMETRIC EQUATIONS KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
Abstract. We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman–Lions–Meyer–Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente’s lemma. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah’s conservation law and give a new regularity proof analogous to H´elein’s for harmonic maps into spheres. Secondly, we prove regularity for solutions to critical systems with nonlocal antisymmetric potentials on the right-hand side. Since the half-harmonic map equation into general target manifolds has this form, as a corollary, we obtain a new proof of the regularity of half-harmonic maps into general target manifolds following closely Rivi`ere’s celebrated argument in the local case. Lastly, the fractional div-curl quantities provide also a new, simpler, proof for H¨ older continuity of W s,n/s -harmonic maps into spheres and we extend this to an argument for W s,n/s -harmonic maps into homogeneous targets. This is an analogue of Strzelecki’s and Toro–Wang’s proof for n-harmonic maps into spheres and homogeneous target manifolds, respectively.
2010 Mathematics Subject Classification. 42B37, 42B30, 35R11, 58E20, 35B65. Key words and phrases. fractional divergence, fractional div-curl lemma, fractional harmonic maps. 1
2
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
Contents 1. Introduction
2
2. Fractional divergence and div-curl lemmas
5
3. Fractional div-curl quantities and half-harmonic maps into spheres
8
4. Fractional div-curl quantities and systems with nonlocal antisymmetric potential and half-harmonic maps into general manifolds 11 5. Fractional div-curl quantities and W s,p-harmonic maps into homogeneous manifolds
16
6. Fractional div-curl estimates: Proof of Theorem 2.1 and Proposition 2.4
19
Appendix A. Nonlocal antisymmetric potential and the optimal gauge: Proof of Proposition 4.2 and Theorem 4.4
25
Appendix B. Euler–Lagrange equations for W s,p -harmonic maps into homogeneous Riemannian manifolds: Proof of Lemma 5.1 29 Appendix C. An integro-differential Triebel–Lizorkin type space
32
References
35
1. Introduction Products of divergence-free and curl-free vector fields, the so-called div-curl-quantities, play a fundamental role in Geometric Analysis. They appear, for example,Vin the theory of compensated compactness in the form of the div-curl Lemma: let L2 ( 1 Rn ) be the L2 -space of 1-forms on Rn , or equivalently the space of vector fields L2 (Rn , Rn ). Given two V V sequences {Fk }k∈N , {Gk }k∈N in L2 ( 1 Rn ) which weakly converge in L2 ( 1 Rn ) to F and G, respectively. In general, there is no reason that the product converges (1.1)
k→∞
Fk · Gk −−−→ F · G in D ′ (Rn ).
If we know, however, that (in distributional sense) div(Fk ) = 0 and curl (Gk ) = 0, or more generally assuming compactness of div(Fk ) and curl (Gk ) in H −1 , then (1.1) indeed holds true. This phenomenon is known as compensated compactness and its theory was developed by Murat and Tartar in the late seventies [27, 28, 45, 46, 47], see also the more recent [7, 9]. In [8] Coifman, Lions, Meyer, and Semmes found a relation between div-curl quantities and the Hardy space H1 (Rn ) (for a definition see Section 6).
FRACTIONAL DIV-CURL QUANTITIES
Theorem 1.1 (Coifman–Lions–Meyer–Semmes). Let F ∈ Lp ( p . Then, if where p ∈ (1, ∞) and p′ = p−1
3
V1
˙ 1,p′ (Rn ) Rn ) and g ∈ W
div F = 0, we have F · dg ∈ H1 (Rn ) with the estimate kF · dgkH1 (Rn ) - kF kLp (V1 Rn ) kdgkLp′ (V1 Rn ) . ′ V This theorem can be applied to all div-curl quantities, since a curl-free G ∈ Lp ( 1 Rn ) can be written as G = dg. Theorem 1.1 had a fundamental impact, in particular, on the regularity theory for such objects as surfaces of prescribed mean curvature and harmonic maps into manifolds [20, 22, 21, 2, 29]. We will detail a few of those results below.
Harmonic maps into spheres. As a first example, consider H´elein’s proof [20] for continuity of harmonic maps from R2 into a round sphere SN −1 ⊂ RN . These are solutions ˙ 1,2 (R2 , RN ) that pointwise a.e. satisfy |u| ≡ 1 and u∈W (1.2)
− ∆ui = ui |du|2
i = 1, . . . , N .
Henceforth, we shall use Einstein’s summation convention. Since |u| ≡ 1 we find 1 uk duk = d|u|2 = 0. 2 Thus, one can rewrite (1.2) as −∆ui = Ωik · duk , where (1.3)
Ωik := ui duk − uk dui .
Shatah discovered in [40], that (1.2) is equivalent to the conservation law (1.4)
div Ωik = 0 for i, k ∈ {1, . . . , N}.
That is, in view of Theorem 1.1, (1.2) actually implies ∆ui ∈ H1 (R2 ) for any i = 1, . . . , N. Then Calderon–Zygmund theory implies that u is continuous, see [39].
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KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
Harmonic maps into general target manifolds. Let N ⊂ RN be a smooth, compact manifold without boundary. Harmonic maps into N are solutions u ∈ W 1,2 (R2 , N ) to − ∆u ⊥ Tu N .
(1.5)
Regularity for harmonic maps from 2-dimensional domains into general manifolds N was proven by H´elein in [21]. Rivi`ere observed in the seminal work [29] that this equation, and in fact the Euler–Lagrange equations of a huge class of conformally invariant variational functionals, have the form (1.6)
− ∆ui = Ωij · ∇uj ,
for i = 1, . . . , N,
V where Ω is an antisymmetric L2 -vector field, Ωij = −Ωji ∈ L2 ( 1 R2 ). By an adaptation of techniques due to Uhlenbeck [52], see also [33], he then constructed a gauge ˙ 1,2 (R2 , SO(N)), such that P ∈W (1.7)
div((∇P )P T − P ΩP T ) = 0.
Then for ΩP := (∇P )P T − P ΩP T , div(Pij ∇uj ) = Pjk (ΩP )ij · ∇uk . Thus, up to a multiplicative Pjk , the right-hand side becomes a div-curl quantity. Continuity of u then follows essentially from Theorem 1.1, cf. [30]. n-harmonic maps into homogeneous target manifolds. H´elein’s regularity argument for harmonic maps into spheres was extended to n-harmonic maps from Rn into spheres SN −1 , see [16, 44, 43]. We follow Strzelecki’s work [43]. Take an n-harmonic map into a ˙ 1,n (Rn , SN −1 ) to sphere SN −1 , i.e., a solution u ∈ W (1.8)
− div(|du|n−2dui ) = ui |du|n,
for i = 1, . . . , N .
This can be rewritten, for Ωij as in (1.3), (1.9)
− div(|du|n−2dui ) = |du|n−2Ωij duj ,
for i = 1, . . . , N.
Again, one can observe that (1.8) is equivalent to div(|du|n−2Ωij ) = 0. Thus the right-hand side of (1.9) is again a div-curl quantity and regularity follows again essentially by Theorem 1.1. Strzelecki’s argument, in turn, was generalized to homogeneous spaces by Toro and Wang [50]. Let us remark here, that the regularity of n-harmonic maps into a general target manifold N is still open, cf. [38].
FRACTIONAL DIV-CURL QUANTITIES
5
Outline of the article. In Section 2 we will introduce a fractional version of divergence and obtain a fractional analogue of Theorem 1.1, see Theorem 2.1 below. As we shall see, this fractional divergence and, in particular, fractional div-curl quantities, appear naturally in the theory of fractional harmonic maps and critical systems with nonlocal antisymmetric potential. We give several examples of consequences of Theorem 2.1 in regularity theory: in Section 3 we consider half-harmonic maps into the sphere. Applications to critical systems with nonlocal antisymmetric potential on the right-hand side will be treated in Section 4. The case of W s,p -harmonic maps into round target manifolds is the subject of Section 5. Finally, in Section 6 we give the proof of the fractional div-curl theorem, Theorem 2.1.
2. Fractional divergence and div-curl lemmas Let us remark, without going into details, that the notion “s-gradient” ds and “sdivergence” divs defined below can be justified by an abstract theory on Dirichlet forms Es,2(f ) := [f ]2W s,2 (Rn ) acting on the Sobolev space W s,2(Rn ), see [23, Examples 4.1]. Here for s ∈ (0, 1), the Gagliardo seminorm [f ]W s,p (Rn ) is given by Z Z p 1 |f (x) − f (y)| dx dy p . [f ]W s,p (Rn ) := |x − y|s |x − y|n Rn Rn We will denote by M(Rn ) the set of all functions measurable with respect to the Lebesgue V measure dx. Furthermore, the space of measurable (off diagonal) vector fields M( 1od Rn ) is the space of functions F : Rn × Rn → R measurable with respect to the measure dx dy . Here “od” stands for “off diagonal”. Observe that we do not require antisymmetry |x−y|n F (x, y) = −F (y, x). On the space of measurable vector fields we define the scalar product. V For two vector fields F, G ∈ M( 1od Rn ) set Z dy . hF, Gi(x) ≡ F · G(x) = F (x, y) G(x, y) |x − y|n Rn In particular we denote kF k2 (x) := and more generally, kF kp (x) := The s-gradient ds : M(Rn ) → M(
Z
p
hF, F i(x),
dy |F (x, y)| |x − y|n Rn
V1
p
p1
.
Rn ) acting on functions g : Rn → R takes the form ^1 g(x) − g(y) n ds g(x, y) := R . ∈ M od |x − y|s od
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KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
In particular, F · ds g(x) =
Z
F (x, y)
Rn
dy g(x) − g(y) . s |x − y| |x − y|n
We will call the operation to the s-gradient ds the s-divergence divs . It maps a vector V1dual n field F ∈ M( od R ) into a function divs F ∈ M(Rn ). Its distributional definition is Z Z dx dy , ϕ ∈ Cc∞ (Rn ). (2.1) divs (F )[ϕ] = F (x, y) ds ϕ(x, y) n |x − y| n n R R In particular, we say that a vector field F is divergence free, divs F = 0, if Z Z ϕ(x) − ϕ(y) dx dy (2.2) F (x, y) = 0 for any ϕ ∈ Cc∞ (Rn ). |x − y|s |x − y|n Rn Rn Moreover, the canonical relation to the fractional Laplacian holds true: divs ds = −(−∆)s , in the sense that Z Z (−∆)s f (x) g(x) dx.
ds f · ds g(x) dx =
Rn
Rn
Here, (−∆)s stands for
s
(−∆) f (x) = P.V.
Z
Rn
dy f (x) − f (y) , 2s |x − y| |x − y|n
or, equivalently, F ((−∆)s f ) (ξ) = c |ξ|2s F (f )(ξ), where F denotes the Fourier transform and c is a multiplicative constant. For p ∈ (1, ∞) the natural Lp -space on vector fields F , is induced by the norm p1 Z Z
p dx dy
kF kp (·) p n . V ≡ kF kLp ( 1od Rn ) := |F (x, y)| L (R ) |x − y|n Rn Rn
Observe that, in particular, we have
[g]W s,p (Rn ) = kds gkLp (V1od Rn ) . Also, if F ∈ Lp (
V1
od
Rn )) then F (x, x) = 0 for almost every x.
Lastly, for Ω ⊆ Rn we denote kF kLp (V1od Ω) :=
Z Z
dx dy |F (x, y)| |x − y|n (Ω×Rn )∪(Rn ×Ω) p
p1
.
A fractional div-curl quantity is then the product F · ds g where F is s-divergence free. For such expressions, we have the fractional counterpart of Theorem 1.1.
FRACTIONAL DIV-CURL QUANTITIES
7
Theorem and Hardy space). Let s ∈ (0, 1), p ∈ (1, ∞). For V1 2.1n (div-curl quantities s,p′ n p F ∈ L ( od R ) and g ∈ W (R ) assume that divs F = 0. Then F · ds g belongs to the Hardy space H1 (Rn ) and we have the estimate kF · ds gkH1 (Rn ) ≤ C kF kLp (V1od Rn ) kds gkLp′ (V1od Rn ) ,
or, equivalently, for any ϕ ∈ Cc∞ (Rn ), Z ϕ F · ds g dx ≤ C [ϕ]BM O(Rn ) kF kLp (V1od Rn ) kds gkLp′ (V1od Rn ) . Rn
Here, C is a uniform constant depending on s, p, and the dimension n.
The definition of BMO and the Hardy space H1 can be found in Section 6 Remark 2.2. It would be interesting to see if the estimate from Theorem 2.1 could also be proved from the harmonic extension to the upper half space, as is possible for classical div-curl structures and many commutators, see [25]. As a first immediate application let us state the following fractional version of Wente’s lemma, see [53, 6, 48]. V Corollary 2.3 (Fractional Wente Lemma). Let s ∈ (0, 1), p ∈ (1, ∞). For F ∈ Lp ( 1od Rn ) ′ and g ∈ W s,p (Rn ) assume that divs F = 0. Moreover, let T [ϕ] be a linear operator such that for some Λ > 0, n
T [ϕ] ≤ Λk(−∆) 4 ϕkL(2,∞) (Rn )
for all ϕ ∈ Cc∞ (Rn ),
where L(2,∞) denotes the weak L2 -space. ˙ n2 ,2 (Rn ) to Then any distributional solution u ∈ W n
(−∆) 2 u = F · ds g + T
in Rn ,
is continuous and if lim|x|→∞ u(x) = 0, then kukL∞ (Rn ) ≤ C kF kLp (V1od Rn ) kds gkLp′ (V1od Rn ) + Λ ,
for a uniform constant C > 0 depending only on s, p, and the dimension n. For weak Lp -spaces L(p,∞) and more generally Lorentz spaces L(p,q) we refer, e.g., to [18, 49]. Proof. This follows from a standard argument, we only give a sketch of the proof: by Theorem 2.1, for any ϕ ∈ Cc∞ (Rn ), Z n n n (−∆) 4 u (−∆) 4 ϕ - [ϕ]BM O kF kLp (V1od Rn ) kds gkLp′ (V1od Rn ) + Λ k(−∆) 4 ϕkL(2,∞) (Rn ) . Rn
n
n
(2,1)
Since [ϕ]BM O - k(−∆) 4 ϕkL(2,∞) (Rn ) , we obtain (−∆) 4 u ∈ Lloc (Rn ). This implies that u is continuous by Sobolev embedding.
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KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
It is also beneficial to have a localized version of Theorem 1.1. A classical version of this result can be found, e.g., in [43, Corollary 3]. V Proposition 2.4 (Localized div-curl estimate). Let F ∈ Lp ( 1od Rn ) be such that divs F = ′ 0 and let g ∈ W s,p (Rn ) for some s ∈ (0, 1), p ∈ (1, ∞). Then, for any ball B(x0 , r) ⊂ Rn and any ϕ ∈ Cc∞ (B(x0 , r)), for a uniform constant Λ > 0 we have Z ϕ F · ds g dx ≤ C [ϕ]BM O + r −n kϕkL1 (Rn ) kF kLp (V1od B(x0 ,Λr)) kds gkLp′ (V1od B(x0 ,Λr)) . Rn
The proofs of Theorem 2.1 and Proposition 2.4 can be found in Section 6. 3. Fractional div-curl quantities and half-harmonic maps into spheres As a first application, let us observe how Theorem 2.1 gives a new, streamlined proof of 1 ˙ 12 ,2 (R) be the the continuity of half-harmonic maps into spheres SN −1 . Let H˙ 2 (R) ≡ W 1 homogeneous Sobolev space of order 21 . By H˙ 2 (R, SN −1 ) we denote the space of maps 1 u ∈ H˙ 2 (R, RN ) such that u(x) ∈ SN −1 for almost every x ∈ R. 1 Half-harmonic maps are solutions u ∈ H˙ 2 (R, SN −1 ) to 1
(−∆) 2 u ⊥ Tu SN −1 . Equivalently, see [26], they satisfy (3.1)
1
(−∆) 2 ui = ui |d 1 u|22 2
in R, for i = 1, . . . , N.
Our first observation is a conservation law, analogous to Shatah’s (1.4), see [40]. 1
Lemma 3.1. A map u ∈ H˙ 2 (R, SN −1 ) is a solution to (3.1) if and only if (3.2)
Ωik (x, y) := ui (x) d 1 uk (x, y) − uk (x) d 1 ui (x, y) 2
2
satisfies (3.3)
div 1 Ωik = 0 2
for all i, k ∈ {1, . . . , N}.
Let us remark that in [13] Da Lio and Rivi`ere obtained an almost-conservation law for horizontal fractional harmonic maps. As a consequence of Lemma 3.1 above we obtain a new proof of 1 Theorem 3.2. Half-harmonic maps, that is solutions u ∈ H˙ 2 (R, SN −1 ) to (3.1) are continuous.
FRACTIONAL DIV-CURL QUANTITIES
9
Regularity for half-harmonic maps was first proved in the pioneering work by Da Lio– Rivi`ere [12]. Another approach was given by Millot–Sire [26] who interpreted the halfharmonic map equation (3.1) as the free boundary condition of a harmonic map U : R2+ → RN ( ∆U i = 0 on R2+ , U =u on R × {0}, 1
observing that then ∂ν U = c (−∆) 2 u on R. Then regularity theory follows from the known regularity results for such free boundary harmonic maps, see [32]. The proof of Theorem 3.2 that we give here follows very closely the original proof for harmonic maps into spheres by H´elein [20]. Proof of Theorem 3.2. As in the local case we rewrite the right-hand side of (3.1). Recall that we use Einstein’s summation convention. With Ωik from (3.2) we find (3.4)
1
(−∆) 2 ui (x) = ui (x) |d 1 u|22(x) ≡ ui (x) hd 1 uk , d 1 uk i(x) = hΩik , d 1 uk i(x) + T (x) 2
2
2
2
where T (x) := hd 1 ui , uk (x) d 1 uk i(x). 2
2
As for T we have Z T (x) ϕ(x) dx - k(−∆) 14 ϕkL(2,∞) (R) (3.5)
for every ϕ ∈ Cc∞ (R).
R
Assuming (3.5) and in view of Lemma 3.1 we have found that the equation (3.4) exhibits a fractional div-curl structure on the right-hand side. Thus, it falls into the realm of the fractional Wente lemma, Corollary 2.3. Hence, Theorem 3.2 is proven once Lemma 3.1 and (3.5) are established.
3.1. Proof of Lemma 3.1. Proof of (3.1) ⇒ (3.3). We compute the fractional divergence div 1 Ωik , see (2.1). For any 2 ϕ ∈ Cc∞ (R), Z Z dx dy Ωik (x, y) d 1 ϕ(x, y) 2 |x − y| R R Z Z dx dy = ui (x) d 1 ϕ(x, y) d 1 uk (x, y) − uk (x) d 1 ϕ(x, y) d 1 ui (x, y) . 2 2 2 2 |x − y| R R Now a simple computation confirms the product rule for d 1 , 2
i
i
i
u (x) d 1 ϕ(x, y) = d 1 (u ϕ)(x, y) − d 1 u (x, y) ϕ(y). 2
2
2
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KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
Thus, dx dy Ωik (x, y)d 1 ϕ(x, y) 2 |x − y| R R Z Z dx dy i k k i = d 1 (u ϕ)(x, y) d 1 u (x, y) − d 1 (u ϕ)(x, y) d 1 u (x, y) 2 2 2 2 |x − y| R R Z Z dx dy . − ϕ(y) d 1 ui (x, y) d 1 uk (x, y) − d 1 uk (x, y) d 1 ui (x, y) 2 2 2 2 |x − y| R R
Z Z
The last line is zero. With (3.1) we find Z Z dx dy i k k i d 1 (u ϕ)(x, y) d 1 u (x, y) − d 1 (u ϕ)(x, y) d 1 u (x, y) 2 2 2 2 |x − y| R R Z = ui (x) uk (x) − uk (x) ui (x) ϕ(x) kd 1 uk22 (x) dx = 0. 2
R
(3.3) is established.
Proof of (3.3) ⇒ (3.1). Equation (3.3) readily implies 1
1
0 = ui (x)(−∆) 2 uj (x) − uj (x)(−∆) 2 ui (x), that is
1
0 = u ∧ (−∆) 2 u, which is equivalent to
1
(−∆) 2 u ⊥ Tu SN −1 . Thus, u is a half-harmonic map.
3.2. Proof of (3.5). For any x, y ∈ R, uk (x) d 1 uk (x, y) = uk (x)
(uk (x) − uk (y))
2
1
|x − y| 2
.
Since |u(x)| ≡ 1 and thus (u(x) − u(y)) · (u(x) + u(y)) = 0, we find uk (x) d 1 uk (x, y) = 2
Thus, Z
1 ϕ(x) u (x) d 1 u · d 1 u (x) dx = 2 2 2 R k
i
k
Z Z R
R
1 |u(x) − u(y)|2 . 2 |x − y| 21
ϕ(x)
ui (x) − ui (y) |u(x) − u(y)|2 dy dx . 1 1 |x − y| |x − y| 2 |x − y| 2
Interchanging x and y, we arrive at Z Z Z 1 |u(x) − u(y)|3 |ϕ(x) − ϕ(y)| k i k ϕ(x) u (x) d 1 u · d 1 u (x) dx ≤ dx dy 2 2 4 R R |x − y|2 R Z Z dx dy . = |d 1 u(x, y))|3 |d 1 ϕ(x, y)| 6 2 |x − y| R R
FRACTIONAL DIV-CURL QUANTITIES
11
From this one obtains by interpolation, Z 1 1 ϕ(x) uk (x) d 1 ui · d 1 uk (x) dx - k((−∆) 6 u)3 kL(2,1) (R) k(−∆) 4 ϕkL(2,∞) (R) . 2
R
2
Moreover, by Sobolev embedding, 1
1
1
k((−∆) 6 u)3 kL(2,1) (R) = k(−∆) 6 uk3L(6,3) (R) - k(−∆) 4 uk3L(2,3) (R) 1
- k(−∆) 4 uk3L2 (R) = [u]3
1
W 2 ,2 (R)
.
This establishes (3.5).
4. Fractional div-curl quantities and systems with nonlocal antisymmetric potential and half-harmonic maps into general manifolds 1
Here we study the regularity theory for a nonlocal analogue of (1.6). Let u ∈ H˙ 2 (R, RN ) be a solution to 1
(−∆) 2 ui = Ωij · d 1 uj , 2 V 1 for some antisymmetric Ωij = −Ωji ∈ L2 ( od R). (4.1)
Observe that the antisymmetric potential Ω is not a pointwise function, but rather acts as a nonlocal operator: one could write the equation above as 1
(−∆) 2 ui = Ωij (uj ), where
Z
Ωij (f )(x) :=
Ωij (x, y)(f (x) − f (y))
dy
3 . |x − y| 2 In [11] Da Lio and Rivi`ere studied the regularizing effects of the equation
R
1
1
(−∆) 2 ui (x) = Ωij (x) (−∆) 4 uj (x), where Ωij = −Ωji ∈ L2 (R) is a function. In [35] the second-named author studied the regularity theory for another class of antisymmetric nonlocal operators, 1
1
(−∆) 2 ui = Ωij H[(−∆) 4 uj ], where H is the Hilbert transform. Here, in the spirit of the celebrated work of Rivi`ere [29], we develop the regularity theory of nonlocal antisymmetric systems of the form (4.1). Namely, we have Theorem 4.1 (Regularity for systems with nonlocal antisymmetric operator). Let u ∈ 1 H˙ 2 (R, RN ) be a weak solution to (4.2)
1
(−∆) 2 ui = Ωij · d 1 uj + R 2
in R.
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KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
V Assume that Ωij = −Ωji ∈ L2 ( 1od R) and that R satisfies, for any ϕ ∈ Cc∞ (R), Z Z Z Z Z dy dx 3 dy dx (4.3) Rϕ |ϕ(x)| |d 1 u(x, y)| + |d 1 u(x, y)|2 |d 1 ϕ(x, y)| . 3 4 2 |x − y| |x − y| R R R R R Then u is H¨older continuous. We postpone the proof and first mention an application. Regularity theory for halfharmonic maps from R into a smooth, compact manifold N ⊂ RN without boundary follows from Theorem 4.1. Indeed, just as the harmonic map equation (1.5) can be brought into the form (1.6), the half-harmonic map equation 1
(−∆) 2 u ⊥ Tu N
(4.4)
in R
can be brought into the form (4.2). 1 Proposition 4.2. Let u ∈ H˙ 2 (R, N ) be a half-harmonic map into a general smooth manifold without boundary N ⊂ RN , i.e., a distributional solution to (4.4). Then u solves (4.2) V for some Ωij = −Ωji ∈ L2 ( 1od R) and some R which satisfies (4.3).
The proof of Proposition 4.2, which we give in Section A.1, follows essentially the local argument used to obtain (1.6). The only difference is that while du is tangential, ds u(x, y) is not — this is why the error term R appears. But since ds u(x, y) is tangential up to a quadratic error, see Lemma A.1, R is benign. Thus, as a corollary of Theorem 4.1 and Proposition 4.2, we obtain
Theorem 4.3 (Da Lio, Rivi`ere [11]). Half-harmonic maps from the line R into a general manifold N are H¨older continuous and in fact smooth. It suffices to show the H¨older continuity. Higher regularity follows from bootstrapping and the growth of the right-hand side – the antisymmetry and the precise right-hand side structure are not relevant. See [35]. 4.1. Proof of Theorem 4.1. As in the local case (1.7), the first step is to find a good 1 gauge P ∈ H˙ 2 (R, SO(N)). V 1 Theorem 4.4. For Ωij = −Ωji ∈ L2 ( 1od R) there exists P ∈ H˙ 2 (R, SO(N)) such that div 1 ΩPij = 0 for all i, j ∈ {1, . . . , N}, 2
where ΩP =
1 d 1 P (x, y) P T (y) + P T (x) − P (x)Ω(x, y)P T (y) − P (y)Ω(x, y)P T (x) . 2 2
This choice of gauge [52], or moving frame [21], is obtained from a minimization argument, cf. [33], and is postponed to Section A.2. Having this good choice for P , we rewrite the equation.
FRACTIONAL DIV-CURL QUANTITIES
13
Lemma 4.5. Let u be a solution to (4.1). For P as in Theorem 4.4 and any ϕ ∈ Cc∞ (R) we have, div 1 (Pik d 1 uk ) = ΩPij · d 1 uℓ Pjℓ + R. 2
2
2
1 2
Here, for some R1 ∈ L ( od R), some g ∈ H˙ (R), and for any ϕ ∈ Cc∞ (R), Z Z Z dx dy Rϕ 1 g(x, y)||d 1 u(x, y)| |ϕ(x)| . |R (x, y)| |d 1 4 4 |x − y| 2
V1 R
R
R
Proof. We have Z Z
dx dy Pik (x) uk (x) − uk (y) (ϕ(x) − ϕ(y)) |x − y|2 R R Z Z dx dy = uk (x) − uk (y) (Pik (x)ϕ(x) − Pik (y)ϕ(y)) |x − y|2 R R Z Z dx dy . + uk (x) − uk (y) (Pik (y) − Pik (x)) ϕ(y) |x − y|2 R R
By (4.1) and since PℓjT (x) Pjm (x) = δℓm , Z Z dx dy uk (x) − uk (y) (Pik (x)ϕ(x) − Pik (y)ϕ(y)) |x − y|2 R R Z Z dx dy = Pik (x) Ωkℓ (x, y) PℓjT (x) Pjm (x)d 1 um (x, y) ϕ(x) . 2 |x − y| R R Also, interchanging x and y and using again PℓjT (x) Pjm (x) = δℓm , Z Z dx dy uk (x) − uk (y) (Pik (y) − Pik (x)) ϕ(y) |x − y|2 R R Z Z dx dy = uk (x) − uk (y) (Pik (y) − Pik (x)) ϕ(x) |x − y|2 R R Z Z dx dy =− d 1 Piℓ (x, y) PℓjT (x) Pjm (x) d 1 um (x, y) ϕ(x) 2 2 |x − y| R R Thus,
dx dy Pik (x) uk (x) − uk (y) (ϕ(x) − ϕ(y)) |x − y|2 R R Z Z dx dy + R, =− ΩPij (x, y) Pjℓ (x) d 1 uℓ (x, y) ϕ(x) 2 |x − y| R R
Z Z
where |R| -
Z Z R
R
dx dy |Ω(x, y)| + |d 1 P (x, y)| |P (x) − P (y)| |d 1 u(x, y)| |ϕ(x)| . 2 2 |x − y|
14
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
From Lemma 4.5 and Theorem 4.4 we see that we have found a div-curl quantity. Now we can apply the localized div-curl estimate, Proposition 2.4. With this, one obtains a decay estimate following the typical procedure for critical nonlocal equations, see [11, 10, 35]. We only give the main ideas of the proof. For any ball B(x0 , r) ⊂ R and any test-function ϕ ∈ Cc∞ (B(x0 , r)) Z Z Z k P ℓ Pik d 1 u d 1 ϕ = Ωij · d 1 u Pjℓ ϕ + Rϕ R
2
2
2
R
R
For any suitably large Λ > 2, u˜ := ηB(x0 ,Λr) (u − (u)B(x0 ,Λr)). for a cutoff-function η ∈ Cc∞ (B(0, 1)) and ηB(x0 ,ρ) (x) := η((x − x0 )/ρ). Then Z Z Z P ℓ P ℓ Ωij · d 1 u Pjℓ ϕ = Ωij · d 1 u˜ Pjℓ ϕ + ΩPij · d 1 uℓ − u˜ℓ Pjℓ ϕ. 2
R
R
2
R
2
By the disjoint support of ϕ and u − u˜, one can show, for some σ > 0 which will change from line to line, Z ∞ X 1 ℓ ℓ P (2k Λ)−σ k(−∆) 4 ukL(2,∞) (B(x0 ,2k Λr) . Ωij · d 1 u − u˜ Pjℓ R
2
k=1
Since ΩPij is divergence free, Z Z ℓ P Ωij · d 1 u˜ Pjℓ ϕ = − ΩPij · d 1 (Pjℓ ϕ) u˜ℓ . R
2
2
R
Now we apply Proposition 2.4, Z ΩPij · d 1 u˜ℓ Pjℓ ϕ R
2
- [˜ u]BM O + (Λr)−1k˜ ukL1 (R) kΩP kL2 (V1od B(x0 ,Λ2 r)) kd 1 (P ϕ) kL2 (V1od B(x0 ,Λ2 r)) . 2
1
By Sobolev embedding, kf kBM O - k(−∆) 4 f kL(2,∞) (R) , which suitably localized gives u]BM O [˜ u]BM O + (Λr)−1k˜ ukL1 - [˜ 1 4
- k(−∆) ukL(2,∞) (B(Λ2 r)) +
∞ X
1
(2k Λ)−σ k(−∆) 4 ukL(2,∞) (B(x0 ,2k Λ2 r) .
k=1
Lastly, 1 1 kd 1 (P ϕ) kL2 (V1od B(x0 ,Λ2 r)) - kP kL∞ (R) + k(−∆) 4 P kL2 (R) k(−∆) 4 ϕkL2 (R) + kϕkL∞ (R) . 2
FRACTIONAL DIV-CURL QUANTITIES
In conclusion, Z Pik d 1 uk d 1 ϕ 2 2
15
R
1
- λ (kϕkL∞ (R) + k(−∆) 4 ϕkL2 (R) ) +
∞ X
1
1
[u]BM O(B(x0 ,Λ2 r)) + k(−∆) 4 ukL(2,∞) (B(x0 ,Λ2 r))
(2k Λ)−σ k(−∆) 4 ukL(2,∞) (B(x0 ,2k Λ2 r) ,
k=1
where
λ = kΩP kL2 (V1od B(x0 ,Λ2 r)) can be chosen arbitrarily small if we restrict our attention to small enough balls, r ≪ 1. By a duality argument, see [35, Lemma 5.18], we find some ϕ ∈ Cc∞ (B(x0 , Λr)), kϕk∞ + 1 k(−∆) 4 ϕkL(2,1) (R) ≤ 1 such that Z 1 1 1 1 k k(−∆) 4 ukL(2,∞) (B(x0 ,r)) - Pik (−∆) 4 u (−∆) 4 ϕ + Λ−σ k(−∆) 4 ukL(2,∞) (B(x0 ,Λ2 r)) R
+
∞ X
1
(2k Λ)−σ k(−∆) 4 ukL(2,∞) (B(x0 ,2k Λ2 r) .
k=1
Now, we have the following commutator-like estimate. Lemma 4.6. For any ϕ ∈ Cc∞ (R) Z Z 1 1 P d 1 u · d 1 ϕ− P (−∆) 4 u (−∆) 4 ϕ 2 2 R R 1 1 1 - k(−∆) 4 ϕkL2 (R) k(−∆) 4 P kL2 (B(x0 ,Λr)) k(−∆) 4 ukL(2,∞) (B(x0 ,Λr)) ∞ X 1 k −σ 4 (2 Λ) k(−∆) ukL(2,∞) (B(x0 ,2k Λr)) . + k=1
Proof. We use the formula ϕ(z) = c
Z
1
1
|x − z| 2 (−∆) 4 ϕ(z) dz
R
and have Z Z Z 1 1 P (x)(u(x) − u(y))(ϕ(x) − ϕ(y)) dx dy − P (z)(−∆) 4 u(z) (−∆) 4 ϕ(z) dz 2 |x − y| R R R Z Z Z (P (x) − P (z))(u(x) − u(y)) |x − z|− 21 − |y − z|− 21 1 dx dy (−∆) 4 ϕ(z) dz. = 2 |x − y| R R R
Now an analysis similar to that of [34, Lemma 6.5, Lemma 6.6.] completes the proof.
16
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
We conclude that, on every ball B(x0 , r) ⊂ R, 1
1
k(−∆) 4 ukL(2,∞) (B(x0 ,r)) - (λ + Λ−σ )k(−∆) 4 ukL(2,∞) (B(x0 ,Λ2 r)) ∞ X 1 (2k Λ)−σ k(−∆) 4 ukL(2,∞) (B(x0 ,2k Λ2 r) . + k=1
This can be seen as a good decay estimate: For any ε > 0 we find a large Λ such that for all small enough radii r ≪ 1 (so that λ is small), 1
1
k(−∆) 4 ukL(2,∞) (B(x0 ,r)) ≤ ε k(−∆) 4 ukL(2,∞) (B(x0 ,Λ2 r)) ∞ X 1 +ε (2k Λ)−σ k(−∆) 4 ukL(2,∞) (B(x0 ,2k Λ2 r) . k=1
Now an iteration argument, see [3, Lemma A.8], implies that there is a θ > 0 such that 1
k(−∆) 4 ukL(2,∞) (B(x0 ,r)) - r θ
for all r > 0.
H¨older continuity of u then follows from Sobolev embedding on Morrey spaces, see [1]. This proves Theorem 4.1. 5. Fractional div-curl quantities and W s,p -harmonic maps into homogeneous manifolds For s ∈ (0, 1), p ∈ (1, ∞) W s,p -harmonic maps from Rn into a manifold N ⊂ RN are ˙ s,p(Rn , N ) to solutions u ∈ W divs (|ds u|p−2ds ui ) ⊥ Tu N . These are exactly the critical points of the energy p Z Z |u(x) − u(y)| dx dy Es,p(u) := ≡ kds ukpLp (Rn ) . s n |x − y| |x − y| Rn Rn The operator divs (|ds u|p−2ds ui ) is often referred to as fractional p-Laplacian (−∆)sp , whose regularity theory has received a lot of attention lately, see, e.g., [14, 24, 5, 36]. 1
The strategy for half-harmonic maps into spheres from Section 3, that is W 2 ,2 -harmonic n maps can be extended to W s, s -harmonic maps, into round targets. Namely we obtain n H¨older regularity for W s, s -harmonic maps from n-dimensional sets into homogeneous spaces. The argument now follows the corresponding classical arguments of Strzelecki [43] and Toro–Wang [50] for n-harmonic maps into spheres and homogeneous manifolds, respectively. First, we rewrite the equation.
FRACTIONAL DIV-CURL QUANTITIES
17
Lemma 5.1 (Euler–Lagrange Equations). Let s ∈ (0, 1) and p ∈ (1, ∞). For any u ∈ W s,p (Rn , N ) which is a W s,p-harmonic map into a homogeneous Riemannian manifold N equivariantly embedded into RN . Then, divs (|ds u|p−2ds ui ) =
(5.1)
m X
|ds u|p−2 Ωα · ds Yαi (u) + R.
α=1
Here, {Yα }m : N → RN is a family of m smooth tangent vector fields on N , Ωα ∈ V1 n α=1 p L ( od R ) and satisfies (5.2) divs |ds u|p−2Ωα = 0 in Rn , for α = 1 . . . , m.
The error term R satisfies, Z Z R ϕ
Rn
Rn
Z
Rn
|d ps′ u(x, y)|p |ds ϕ(x, y)|
dy dx . |x − y|n
The proof is a direct fractional analogue of the arguments in [22, 43, 50], we postpone it to Section B. From Lemma 5.1 and Theorem 2.1 we obtain the following regularity theorem. This genern alizes the second-named author’s regularity result for W s, s -harmonic maps into spheres [34] to homogeneous target manifolds. Let us stress that even for round spheres our argument is much simpler. Theorem 5.2. Let s ∈ (0, 1), p = ns ∈ [2, ∞) and u ∈ W s,p (Rn , N ) be a W s,p-harmonic map, where N is a homogeneous Riemannian manifold equivariantly embedded into RN . Then u is H¨older continuous. Sketch of the proof. Let B(x0 , r) ⊂ Rn be a ball centered in x0 and r > 0. For a typical cutoff function η ∈ Cc∞ (B(0, 2)), η ≡ 1 on B(0, 1), let ηB(x0 ,r) (x) = η((x − x0 )/r). Define the test-function ϕ by ϕ := ηB(x0 ,r) (u − (u)B(x0 ,r) ), where (u)B(x0 ,r) denotes the mean value of u on B(x0 , r). Observe that for any Λ > 4, we have the following estimates [ϕ]W s,p (Rn ) - kds ukLp (V1od B(x0 ,Λr)) and since p = ns , [ϕ]BM O + r −n kϕkL1 (Rn ) - kds ukLp (V1od B(x0 ,Λr)) . We also denote by [u]pW s,p (B(x0 ,r))
:=
Z
B(x0 ,r)
|u(x) − u(y)|p dx dy. n+sp B(x0 ,r) |x − y|
Z
18
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
By the usual cutoff-arguments, see for example [34, Lemma 4.1] we find for any ε > 0 a constant Cε > 0 such that Z p [u]W s,p (B(x0 ,r)) ≤ |ds u|p−2ds u · ds ϕ(y) dy Rn + ε [u]pW s,p (B(x0 ,Λr)) + Cε [u]pW s,p (B(x0 ,Λr)) − [u]pW s,p (B(x0 ,r)) +
∞ X
(2k Λ)−σ [u]pW s,p (B(x0 ,2k Λ2 r)) .
k=1
Here σ > 0 is a constant that may vary from line to line. For the first term, we use the equation (5.1) Z Z Z p−2 p−2 i |ds u| ds u · ds ϕ = |ds u| Ωα · ds Yα (u) ϕ + Rn
Rn
Rϕ.
Rn
For the error term R the usual cutoff arguments and Sobolev inequality lead to Z Rϕ - kds ukpLp (V1 B(x ,Λr)) kds ϕkLp (V1od B(x0 ,Λr)) . od
Rn
0
Moreover, we use the div-curl structure induced by (5.2) and Proposition 2.4, to obtain Z |ds u|p−2Ωα · ds Yαi (u) ϕ n R - k|ds u|p−2ΩkLp′ (V1od B(x0 ,Λr)) kds ukLp (V1od B(x0 ,Λr)) [ϕ]BM O + r −n kϕkL1 (Rn ) .
Thus, with the estimate for ϕ, Z |ds u|p−2ds u · ds ϕ - kΩkLp (V1od B(Λr)) + kds ukLp (V1od B(Λr)) kds ukpLp (V1
od
Rn
B(Λr))
.
For any ε > 0, by absolute continuity of integrals, we find R > 0 small enough so that for any r ∈ (0, Λ−1R) we have kΩkLp (V1od B(x0 ,Λr)) + kds ukLp (V1od B(x0 ,Λr)) < ε. Also,
kds ukpLp (V1 B(x ,Λr)) 0 od
-
[u]pW s,p (B(x0 ,Λ2 r))
+
∞ X k=1
Consequently we showed
2k Λ
−σ
[u]pW s,p (B(x0 ,2k Λ2 r)) .
[u]pW s,p (B(x0 ,r)) ≤ 2ε [u]pW s,p (B(x0 ,Λ2 r)) + Cε [u]pW s,p (B(x0 ,Λr)) − [u]pW s,p (B(x0 ,r)) +C
∞ X k=1
2k Λ
−σ
[u]pW s,p (B(x0 ,2k Λ2 r)) .
FRACTIONAL DIV-CURL QUANTITIES
19
Absorbing Cε [u]pW s,p (B(r,x0 )) to the left-hand side, choosing Λ large enough for τ = we find ∞ X −σ p p p 2k Λ [u]W s,p (B(x0 ,2k Λ2 r)) . [u]W s,p (B(x0 ,r)) ≤ τ [u]W s,p (B(x0 ,Λ2 r)) +
Cε +2ε Cε +1
0 such that for any r > 0 [u]W s,p (B(x0 ,r)) - r θ [u]W s,p (Rn ) . Now since p =
n s
by Sobolev inequality on Morrey spaces, see [1], u is H¨older continuous.
6. Fractional div-curl estimates: Proof of Theorem 2.1 and Proposition 2.4 R Fix a smooth nonnegative bump function κ ∈ Cc∞ (B(0, 1)) such that Rn κ = 1. Denote by κt (x) := t−n κ(x/t). The Hardy space H1 (Rn ) is the space of all functions f ∈ L1 (Rn ) such that (6.1)
kf kH1 (Rn ) := k sup |κt ∗ f |kL1(Rn ) < ∞. t>0
The space BMO is given through the seminorm Z −n (6.2) [f ]BM O := sup t t>0, x∈Rn
|f − (f )B(x,t) |.
B(x,t)
Denote by M the Hardy–Littlewood maximal function Z |f |. Mf (x) := sup t>0
B(x,t)
It is well known (see, e.g., [4, 19]), that for any p ∈ [1, ∞), 1
sup t−1 |f (x) − (f )B(x,t) | - (M|df |p) (x) p . t>0
The following is a nonlocal version of this fact. Lemma 6.1. Let s ∈ (0, 1), then for any p ∈ [1, ∞), t > 0, Z p1 |f (x) − f (y)|p −s . t |f (x) − (f )B(x,t) | dy |x − y|n+sp B(x,t) Proof. This can be checked by a direct computation: For any t > 0, Z −s −n−s |f (x) − f (y)| dy. (6.3) t |f (x) − (f )B(x,t) | - t B(x,t)
20
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
By Jensen’s inequality for p ≥ 1, then (6.3) can be further estimated by p1 Z p −s −n−sp |f (x) − f (y)| dy t |f (x) − (f )B(x,t) | - t B(x,t)
=
Z
B(x,t)
|f (x) − f (y)|p |x − y|n+sp
|x − y| t
n+sp
dy
! p1
.
Now the claim follows, since |x − y| ≤ 2t for any y ∈ B(x, t).
Now we give the proof of Theorem 2.1, for which we adapt the argument of Coifman– Lions–Meyer–Semmes [8, Lemma II.1]. Proof of Theorem 2.1. Set Γ(t, x) := |κt ∗ (F · ds g)(x)|. We will show k sup Γ(t, ·)kL1(Rn ) - kF kLp (V1od Rn ) kds gkLp′ (V1od Rn ) ,
(6.4)
t>0
which in view of (6.1) implies the claim of Theorem 2.1. For t > 0 and x ∈ Rn , since divs F = 0 we have Γ(t, x) = κt ∗ (F · ds g)(x) Z Z dz dy = κt (x − y)F (z, y) (g(z) − g(y)) |z − y|n+s n n ZR ZR = F (z, y) (κt (x − y) − κt (x − z)) (g(z) − (g)B(x,2t) ) Rn
Rn
dz dy . |z − y|n+s
By the Lipschitz continuity of κ we find, Z Z dz dy −n−1 |g(z) − (g)B(x,2t) | |F (z, y)| |Γ(t, x)| - t |z − y|n+s−1 B(x,2t) B(x,2t) Z Z dz dy |g(z) − (g)B(x,2t) | |F (z, y)| + t−n |z − y|n+s (6.5) B(x,t) Rn \B(x,2t) Z Z dz dy |g(z) − (g)B(x,2t) | |F (z, y)| + t−n |z − y|n+s Rn \B(x,2t) B(x,t) =: I(t, x) + II(t, x) + III(t, x).
As for I(t, x), by H¨older inequality we obtain, I(t, x) −n−1
≤t
Z
B(x,2t)
Z
B(x,2t)
′
|g(z) − (g)B(x,2t) |p dy |z − y|n+p′(s−1)
! 1′ Z p
dy |F (z, y)| |z − y|n B(x,2t) p
p1
dz.
FRACTIONAL DIV-CURL QUANTITIES
21
Since s < 1, we can integrate the first term in y and have Z Z −n−s |g(z) − (g)B(x,2t) | |F (z, y)|p (6.6) I(t, x) - t B(x,2t)
B(x,2t)
By H¨older inequality for some q > 1, which will be specified below, Z q1 Z Z q −n−s |g(z) − (g)B(x,2t) | dz -t |F (z, y)|p B(x,2t)
−n( n+sq ) nq
-t
Z
Rn
B(x,2t)
q
|g(z) − (g)B(x,2t) | dz B(x,2t)
where we recall our notation kF kp (y) :=
Z
q1
′ MkF kqp (x)
1′
,
p1
.
dz |F (z, y)|p |z − y|n Rn
q
dy |z − y|n
p1
dy |z − y|n
qp′
dz.
1′ q
dz
Whenever q is so that q > p′
(6.7)
we can apply Lemma C.4, for q2 := Z
q
|g(z) − (g)B(x,2t) | dz B(x,2t)
and q >
nq n+sq
q1
n n−s
> 1 and have
-
Z
B(x,Λt)
Z
′
Rn
|g(z) − g(y)|p dz |z − y|n+p′s
qp2′
! q1
2
dy
,
for some constant Λ > 1. Recall again, that kds gkp′ (y) :=
Z
|ds g(y, z)|
Rn
we obtain I(t, x) -
1
Mkds gkqp2′ (x) q2
p′
dz |z − y|n
1′ p
,
1′ q q′ . MkF kp (x)
Thus, from H¨older inequality we get Z Z 1′ q1 q q2 q′ 2 sup I(t, x) dx Mkds gkp′ (x) dx MkF kp (x) Rn t>0
(6.8)
Rn
≤
Z
p′
Rn
Mkds gkqp2′ (x) q2
1′ Z p dx
Rn
′ MkF kqp (x)
1/q
′ 1/q ′ ≤ Mkds gkqp2′ Lp′ 2/q2 (Rn ) MkF kqp Lp/q′ (Rn ) .
p′ q
p1 dx
22
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
Now to apply the Maximal Theorem, see [42, Theorem 1(c), p.13], we choose p/q ′ > 1 and p′ /q2 > 1. The latter can be always achieved for a q satisfying (6.7) by taking q ′ close enough to p′ . Therefore, Z
1/q2
′ 1/q ′ sup I(t, x) dx - kds gkqp2′ Lp′ /q2 (Rn ) kF kqp Lp/q′ (Rn ) Rn t>0
= kds gkLp′ (V1od Rn ) kF kLp (V1od Rn ) .
As for II(t, x), applying twice H¨older inequality and using the definition of the maximal function, we obtain II(t, x) −n
-t
Z
B(x,t)
Z
′
Rn \B(x,2t)
|g(z) − (g)B(x,t) |p dz |z − y|n+sp′
! 1′ Z p
|F (z, y)|p dz |z − y|n
Rn \B(x,2t)
1p
dy
! 1′ Z p
Z 1p |g(z) − (g)B(x,t) | |F (z, y)|p dy dz dz -t sup n |z − y|n+sp′ y∈B(x,t) Rn \B(x,2t) |z − y| B(x,t) Rn \B(x,2t) ! 1′ Z p p′ |g(z) − (g)B(x,t) | MkF kp(x) - sup dz |z − y|n+sp′ y∈B(x,t) Rn \B(x,2t) ! 1′ Z ′ p |g(z) − (g)B(x,t) |p MkF kp(x). ≈ dz |z − x|n+sp′ Rn \B(x,2t) −n
We estimate Z
p′
Rn \B(x,2t)
-
≈
p′
Z
Z Z
|g(z) − (g)B(x,t) | dz |z − x|n+sp′ ′
Rn \B(x,2t)
|g(z) − g(x)|p dz |z − x|n+sp′
Rn \B(x,2t)
|g(z) − g(x)|p dz |z − x|n+sp′
′
! 1′ p
p1′
+
p1′
+ t−s |g(x) − (g)B(x,t) |.
′
Z
Rn \B(x,2t)
|g(x) − (g)B(x,t) |p dz |z − x|n+sp′
Thus, by Lemma 6.1, (6.9)
′
Z
Rn \B(x,2t)
|g(z) − (g)B(x,t) |p dz |z − x|n+sp′
! 1′ p
-
Z
Rn
′
|g(z) − g(x)|p dz |z − x|n+sp′
Consequently, we arrive at sup II(t, x) - MkF kp (x) t>0
Z
Rn
′
|g(z) − g(x)|p dz |z − x|n+sp′
p1′
.
p1′
.
! 1′ p
FRACTIONAL DIV-CURL QUANTITIES
23
Integrating this, applying H¨older and then the maximal inequality, similarly as in (6.8), we have shown that k sup II(t, x)kL1 (Rn ) - kds gkLp′ (V1od Rn ) kF kLp (V1od Rn ) . t>0
Now we estimate III(t, x). By H¨older inequality we find III(t, x) −n
=t
−n
-t
Z
B(x,t)
Z
Rn \B(x,2t)
Z
-t
Z
′
|g(z) − (g)B(x,2t) |p dy |z − y|n+sp′ Rn \B(x,2t) Z |g(z) − (g)B(x,2t) | |F (z, y)|p
Z
B(x,t)
−n−s
dz dy |z − y|n+s ! 1′ Z p
|g(z) − (g)B(x,2t) | |F (z, y)|
B(x,t)
Rn
dy |F (z, y)| |z − y|n Rn \B(x,2t) 1p dy dz. |z − y|n p
p1
dz
Enlarging the set on which we integrate to B(x, 2t) we obtain a similar estimate as for I (see (6.6)). Therefore, we have Z sup III(t, x) dx - kds gkLp′ (V1od Rn ) kF kLp (V1od Rn ) . Rn t>0
Now (6.4) is established and the proof is complete.
Proof of Proposition 2.4. By a rescaling argument, we may assume that r = 1 and x0 = 0. We denote by B(r) balls of radius r > 0 centered at the origin. We need two things. Firstly, a simple H¨older inequality yields (6.10)
kF · ds gkL1 (B(10)) - kF kLp (V1od B(10)) kds gkLp′ (V1od B(10)) .
Secondly, let ηB(4) be a nonnegative bump function constantly one on B(4) and vanishing on Rn \B(5). 1 We claim that ηB(4) F · ds g ∈ Hloc (B(8)). More precisely, set
Γ(t, x) := |κt ∗ (ηB(4) F · ds g)(x)|, then we claim (6.11)
k sup |Γ(t, x)|kL1 (B(8)) - kF kLp (V1od B(10)) kds gkLp′ (V1od B(10)) . t∈(0,1)
Assume that (6.11) is proven. Let ηB(3) be another nonnegative bump function constantly one on B(3) and zero on Rn \B(4). In particular, ηB(3) F · ds g = ηB(3) (ηB(4) F · ds g). For λ :=
R
ηB(3) F · ds g R ηB(3)
24
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
we have by [39, Proposition 1.92], kηB(3) (F · ds g − λ)kH1 (Rn ) - kF · ds gkL1 (B(8)) + k sup |Γ(t, x)|kL1 (B(8)) . t∈(0,1)
In particular, for any ϕ ∈ Cc∞ (B(1)), Z - [ϕ]BM O kηB(3) (F · ds g − λ)kH1 (Rn ) + |λ| kϕkL1 (Rn ) . ϕ F · d g s Rn
That is, once (6.11) is established, we have Z V V n ϕ F · ds g - [ϕ]BM O + kϕkL1 (Rn ) kF kLp ( 1od B(10)) kds gkLp′ ( 1od B(10)) . R
It remains to prove (6.11).
We follow the strategy of the proof of Theorem 2.1 above. The only difference is that we have to take into account the ηB(4) -term. Since F is divergence free, Γ(t, x) Z Z dz dy = κt (x − y) ηB(4) (y) F (z, y) (g(z) − g(y)) |y − z|n+s n n ZR ZR dz dy = F (z, y) ηB(4) (y) κt (x − y) − ηB(4) (z) κt (x − z) (g(z) − (g)B(x,2t) ) |y − z|n+s n n ZR ZR dz dy = ηB(4) (z) F (z, y) (κt (x − y) − κt (x − z)) (g(z) − (g)B(x,2t) ) |y − z|n+s Rn Rn Z Z dz dy + F (z, y) ηB(4) (y) − ηB(4) (z) κt (x − y) (g(z) − (g)B(x,2t) ) |y − z|n+s Rn Rn =: I(t, x) + II(t, x). The first term can be treated similarly as Γ(t, x) in the proof of Theorem 2.1, and hence we obtain Z sup |I(t, x)| - kF kLp (V1od B(10)) kds gkLp′ (V1od B(10)) . B(8) t∈(0,1)
For the second term, using the Lipschitz continuity of ηB(4) , for t ∈ (0, 1) and x ∈ B(1), we get Z Z dz dy −n |F (z, y)| |g(z) − (g)B(x,2t) | II(t, x) - t |y − z|n+s−1 B(x,2t) B(x,2t) Z Z dz dy |g(z) − (g)B(x,2t) | |F (z, y)| + t−n |y − z|n+s B(x,t) Rn \B(x,2t) =: II1 (t, x) + II2 (t, x).
FRACTIONAL DIV-CURL QUANTITIES
25
To estimate II1 (t, x) we proceed as with I(t, x) in the proof of Theorem 2.1, (in comparison to (6.5) we gain a t), and we obtain k sup II1 (t, ·)kL1(B(8)) - kF kLp (V1od B(10)) kds gkLp′ (V1od B(10)) . t∈(0,1)
For II2 (t, x) we follow the estimate of II(t, x) in the proof of Theorem 2.1, |II2 (t, x)| -
Z
′
Rn \B(x,2t)
|g(z) − (g)B(x,t) |p dz |z − x|n+sp′
! 1′ p
Mt p. One could think that X˙ p,q = F˙ p,q , however we were not able to immediately prove (or disprove) this.
Proof. In the arxiv-version of [34], in Proposition 8.6, one can find the (easy) proof of Z q1 q |f (x) − f (y)| 2js |∆j f (x)| dy . |x − y|n+sq Rn
Here, ∆j is the j-th Littlewood–Paley projection operator, see [18, 17]. Thus, ! p1 pq p Z p1 Z Z q |f (x) − f (y)| dx . dy sup 2js |∆j f (x)| ≡ kf kF˙p,∞ dx s |x − y|n+sq n j∈Z R Rn Rn s For the Triebel–Lizorkin spaces F˙ p,∞ we have a Sobolev embedding (see, e.g., [51, Theorem 2.7.1 (ii)]) F˙ ps11,q1 (Rn ) ֒→ F˙ ps22,q2 (Rn )
FRACTIONAL DIV-CURL QUANTITIES
33
holds whenever s1 > s2 and for any q1 , q2 ∈ [1, ∞] if p1 , p2 ∈ (1, ∞) are such that n n = s2 − . s1 − p1 p2 From this, (C.1), and Proposition C.1 we have in particular the following Sobolev-type inequality, cf. [34, Theorem 1.6]. Proposition C.3. Let 0 ≤ s2 < s1 < 1 and p1 , p2 , q1 ∈ (1, ∞) so that n n s1 − = s2 − . p1 p2 Then ! p1 pq1 Z Z 1 1 s2 |f (x) − f (y)|q1 . dx dy (C.2) k(−∆) 2 f kLp2 (Rn ) n+s1 q1 Rn Rn |x − y| From Proposition C.3 we also obtain the following localized version. Lemma C.4. Let s ∈ (0, 1) and p, p2 ∈ (1, ∞). For any q ∈ (1, p2 ), if n n (C.3) s− =− p p2 then there is some Λ > 0 so that for any ball B(x0 , ρ) we have ! 1p Z pq Z |f (x) − f (y)|q kf − (f )B(x0 ,ρ) kLp2 (B(x0 ,ρ)) ≤ C dy . dx n+sq B(x0 ,Λρ) B(x0 ,Λρ) |x − y| Here (f )B(x0 ,ρ) denotes the mean value. The constant C depends only on s, p, p2, q and the dimension. Proof. By translation invariance and scaling we may assume that ρ = 1 and x0 = 0. We denote balls centered at the origin by B(r) := B(0, r), for r > 0. Let η ∈ Cc∞ (B(2)) be a typical cutoff function, η ≡ 1 on B(1), k∇ηkL∞ - 1. We apply Proposition C.3 to u(x) := η(x)(f (x) − (f )B(1) ). Then, ! p1 pq Z Z |η(x)(f (x) − (f )B(1) ) − η(y)(f (y) − (f )B(1) )|q kukLp2 (Rn ) dy dx |x − y|n+sq Rn Rn - I + II + III, where I :=
Z
B(Λ)
Z
B(Λ)
! p1 pq |η(x)(f (x) − (f )B(1) ) − η(y)(f (y) − (f )B(1) )|q , dx dy |x − y|n+sq
34
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
II :=
Z
B(Λ)
Z
Rn \B(Λ)
! p1 pq |η(y)(f (y) − (f )B(1) )|q , dx dy |x − y|n+sq
and III :=
Z
Rn \B(Λ)
As for I we estimate Z I-
Z
B(2)
! p1 pq |η(x)(f (x) − (f )B(1) )|q . dx dy |x − y|n+sq
! p1 pq |(η(x) − η(y))(f (x) − f (y))|q dy dx |x − y|n+sq B(Λ) B(Λ) ! p1 pq Z Z |η(y)(f (x) − f (y))|q dy dx + |x − y|n+sq B(Λ) B(Λ) +
Z
Z
B(Λ)
Z
! p1 pq |(η(x) − η(y))(f (y) − (f )B(1) )|q dx dy . |x − y|n+sq
B(Λ)
Regrouping and using the Lipschitz continuity of η we find ! p1 pq Z Z |f (x) − f (y)|q dy dx I|x − y|n+sq B(Λ) B(Λ) +
! p1 pq |f (y) − (f )B(1) |q . dx dy |x − y|n+(s−1)q
Z
Z
B(Λ)
B(Λ)
Since s < 1 we can integrate in x and have ! p1 Z pq Z q |f (y) − (f )B(1) | dx dy n+(s−1)q B(Λ) B(Λ) |x − y| Z p1 p |f (y) − (f )B(1) | dy B(Λ)
-
-
Z
B(Λ)
Z
B(1)
Z
Z
Z
Z
B(Λ)
B(1)
p p1 |f (y) − f (x)| dx dy
! p1 pq |f (y) − f (x)|q dy . dx |x − y|n+sq
That is, I-
B(Λ)
B(Λ)
! p1 pq |f (x) − f (y)|q dy . dx |x − y|n+sq
FRACTIONAL DIV-CURL QUANTITIES
As for II, we integrate in x and have Z p1 p II |f (y) − (f )B(1) | dy B(2)
Z
B(2)
Z
B(1)
35
! p1 pq |f (x) − f (y)|q dy . dx |x − y|n+sq
Finally we estimate III. For Λ > 4, for any x ∈ B(2) and any y ∈ Rn \ B(Λ) we have |x − y| ≥ 21 |y| and we get ! p1 p Z Z p
|y|−(n+sq) q
III -
|η(x)(f (x) − (f )B(1) )|q dx
Rn \B(Λ)
q
dy
.
B(2)
Now, since q ∈ (1, p2 ) we have −(n + sq) pq < −n. Hence, p1 Z −(n+sq) pq - Λ−σ , dy |y| Rn \B(Λ)
for σ :=
n q
+ − np + s > 0. Thus, by the definition of u, Z 1q q −σ |η(x)(f (x) − (f )B(1) )| dx III - Λ B(2)
-Λ
−σ
kukLq (B(2)) - Λ−σ kukLp2 (Rn ) .
In the last inequality we used the fact that q < p2 and u is supported in B(2). We have thus shown, for any Λ > 4, ! p1 pq Z Z q |f (x) − f (y)| . dy dx kukLp2 (Rn ) - Λ−σ kukLp2 (Rn ) + |x − y|n+sq B(Λ) B(Λ) For Λ sufficiently large we can absorb Λ−σ kukLp2 (Rn ) into the left-hand side. This finishes the proof of Lemma C.4. Acknowledgment. We would like to thank M. Hinz for pointing out literature regarding Dirichlet forms and the subsequent gradients and divergence. Both authors are supported by the German Research Foundation (DFG) through grant no. SCHI-1257-3-1. A.S. receives funding from the Daimler and Benz foundation. A.S. is Heisenberg fellow. References [1] D. R. Adams. A note on Riesz potentials. Duke Math. J., 42(4):765–778, 1975. [2] F. Bethuel. Un r´esultat de r´egularit´e pour les solutions de l’´equation de surfaces `a courbure moyenne prescrite. C. R. Acad. Sci. Paris S´er. I Math., 314(13):1003–1007, 1992. [3] S. Blatt, Ph. Reiter, and A. Schikorra. Harmonic analysis meets critical knots. Critical points of the M¨ obius energy are smooth. Trans. Amer. Math. Soc., 368(9):6391–6438, 2016.
36
KATARZYNA MAZOWIECKA AND ARMIN SCHIKORRA
[4] B. Bojarski and P. Hajlasz. Pointwise inequalities for Sobolev functions and some applications. Studia Math., 106(1):77–92, 1993. [5] L. Brasco and E. Lindgren. Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case. Adv. Math., 304:300–354, 2017. [6] H. Brezis and J.-M. Coron. Multiple solutions of H-systems and Rellich’s conjecture. Comm. Pure Appl. Math., 37(2): 149–187, 1984. [7] M. Briane, J. Casado-D´ıaz, and F. Murat. The div-curl lemma “trente ans apr`es”: an extension and an application to the G-convergence of unbounded monotone operators. J. Math. Pures Appl. (9), 91(5):476–494, 2009. [8] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes. Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9), 72(3):247–286, 1993. [9] S. Conti, G. Dolzmann, and S. M¨ uller. The div-curl lemma for sequences whose divergence and curl are compact in W −1,1 . C. R. Math. Acad. Sci. Paris, 349(3-4):175–178, 2011. [10] F. Da Lio. Fractional harmonic maps into manifolds in odd dimension n > 1. Calc. Var. Partial Differential Equations, 48(3-4):421–445, 2013. [11] F. Da Lio and T. Rivi`ere. Sub-criticality of non-local Schr¨odinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math., 227(3):1300–1348, 2011. [12] F. Da Lio and T. Rivi`ere. Three-term commutator estimates and the regularity of 12 -harmonic maps into spheres. Anal. PDE, 4(1):149–190, 2011. [13] F. Da Lio and T. Rivi`ere. Horizontal α-harmonic maps. Preprint, arXiv:1604.05461, 2016. [14] A. Di Castro, T. Kuusi, and G. Palatucci. Local behavior of fractional p-minimizers. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 33(5):1279–1299, 2016. [15] A. Freire. Global weak solutions of the wave map system to compact homogeneous spaces. Manuscripta Math., 91(4):525–533, 1996. [16] M. Fuchs. The blow-up of p-harmonic maps. Manuscripta Math., 81(1-2):89–94, 1993. [17] L. Grafakos. Modern Fourier analysis, volume 250 of Graduate Texts in Mathematics. Springer, New York, second edition, 2009. [18] L. Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014. [19] P. Hajlasz. Sobolev spaces on an arbitrary metric space. Potential Anal., 5(4):403–415, 1996. [20] F. H´elein. R´egularit´e des applications faiblement harmoniques entre une surface et une sph`ere. C.R. Acad. Sci. Paris 311, S´erie I, pages 519–524, 1990. [21] F. H´elein. R´egularit´e des applications faiblement harmoniques entre une surface et une vari´et´e riemannienne. C.R. Acad. Sci. Paris 312, S´erie I, pages 591–596, 1991. [22] F. H´elein. Regularity of weakly harmonic maps from a surface into a manifold with symmetries. Manuscripta Math., 70(2):203–218, 1991. [23] M. Hinz. Magnetic energies and Fey-Kac-Itˆ o formulas for symmetric Markov processes. Stoch. Anal. Appl., 33(6):1020–1049, 2015. [24] T. Kuusi, G. Mingione, and Y. Sire. Nonlocal equations with measure data. Comm. Math. Phys., 337(3):1317–1368, 2015. [25] E. Lenzmann and A. Schikorra. Sharp commutator estimates via harmonic extensions. ArXiv: 1609.08547, September 2016. [26] V. Millot and Y. Sire. On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres. Arch. Ration. Mech. Anal., 215(1):125–210, 2015. [27] F. Murat. Compacit´e par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5(3):489–507, 1978. [28] F. Murat. Compacit´e par compensation: condition n´ecessaire et suffisante de continuit´e faible sous une hypoth`ese de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8(1):69–102, 1981. [29] T. Rivi`ere. Conservation laws for conformally invariant variational problems. Invent. Math., 168(1):1– 22, 2007.
FRACTIONAL DIV-CURL QUANTITIES
37
[30] T. Rivi`ere and M. Struwe. Partial regularity for harmonic maps and related problems. Comm. Pure Appl. Math., 61(4):451–463, 2008. [31] T. Runst and W. Sickel. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, volume 3 of De Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 1996. [32] Ch. Scheven. Partial regularity for stationary harmonic maps at a free boundary. Math. Z., 253(1):135– 157, 2006. [33] A. Schikorra. A remark on gauge transformations and the moving frame method. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 27(2):503–515, 2010. [34] A. Schikorra. Integro-differential harmonic maps into spheres. Comm. Partial Differential Equations, 40(3):506–539, 2015. [35] A. Schikorra. ε-regularity for systems involving non-local, antisymmetric operators. Calc. Var. Partial Differential Equations, 54(4):3531–3570, 2015. [36] A. Schikorra. Nonlinear commutators for the fractional p-Laplacian and applications. Math. Ann., 366(1-2):695–720, 2016. [37] A. Schikorra, Y. Sire, and C. Wang. Weak solutions of geometric flows associated to integro-differential harmonic maps. 2015. [38] A. Schikorra and P. Strzelecki. Invitation to H-systems in higher dimensions: known results, new facts, and related open problems. Preprint, arXiv:1606.07891, 2016. [39] S. Semmes. A primer on Hardy spaces, and some remarks on a theorem of Evans and M¨ uller. Comm. Partial Differential Equations, 19(1-2):277–319, 1994. [40] J. Shatah. Weak solutions and development of singularities of the SU(2) σ-model. Comm. Pure Appl. Math., 41(4):459–469, 1988. [41] L. Simon. Theorems on regularity and singularity of energy minimizing maps. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, 1996. Based on lecture notes by Norbert Hungerb¨ uhler. [42] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [43] P. Strzelecki. Regularity of p-harmonic maps from the p-dimensional ball into a sphere. Manuscripta Math., 82(3-4):407–415, 1994. [44] H. Takeuchi. Some conformal properties of p-harmonic maps and a regularity for sphere-valued pharmonic maps. J. Math. Soc. Japan, 46(2):217–234, 1994. [45] L. Tartar. Une nouvelle m´ethode de r´esolution d’´equations aux d´eriv´ees partielles non lin´eaires. In Journ´ees d’Analyse Non Lin´eaire (Proc. Conf., Besan¸con, 1977), volume 665 of Lecture Notes in Math., pages 228–241. Springer, Berlin, 1978. [46] L. Tartar. Compensated compactness and applications to partial differential equations. In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, volume 39 of Res. Notes in Math., pages 136–212. Pitman, Boston, Mass.-London, 1979. [47] L. Tartar. The compensated compactness method applied to systems of conservation laws. In Systems of nonlinear partial differential equations (Oxford, 1982), volume 111 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 263–285. Reidel, Dordrecht, 1983. [48] L. Tartar. Remarks on Oscillations and Stokes’ Equation. Lecture Notes in Physics, 230, macroscopic Modelling of Turbulent Flows, Proceedings, Sophia-Antipolis, France, pages 24–31, 1984. [49] L. Tartar. An introduction to Sobolev spaces and interpolation spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin; UMI, Bologna, 2007. [50] T. Toro and C. Wang. Compactness properties of weakly p-harmonic maps into homogeneous spaces. Indiana Univ. Math. J., 44(1):87–113, 1995. [51] H. Triebel. Theory of function spaces, volume 78 of Monographs in Mathematics. Birkh¨auser Verlag, Basel, 1983. [52] K. K. Uhlenbeck. Connections with Lp bounds on curvature. Comm. Math. Phys., 83(1):31–42, 1982.
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[53] H. C. Wente. An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl., 26: 318–344, 1969. ¨r Reine Mathematik, Albert(Katarzyna Mazowiecka) Mathematisches Institut, Abt. fu ¨t, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany Ludwigs-Universita & Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland E-mail address:
[email protected] ¨r Reine Mathematik, Albert-Ludwigs(Armin Schikorra) Mathematisches Institut, Abt. fu ¨t, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany Universita E-mail address:
[email protected]
