A fractional version of Rivière's GL(N)-gauge
Francesca Da Lio, Katarzyna Mazowiecka, Armin Schikorra, Lifeng Wang
aa r X i v : . [ m a t h . A P ] J a n A FRACTIONAL VERSION OF RIVI `ERE’S GL(N)-GAUGE
FRANCESCA DA LIO, KATARZYNA MAZOWIECKA, ARMIN SCHIKORRA, AND LIFENG WANG
Abstract.
We prove that for antisymmetric vectorfield Ω with small L -norm thereexists a gauge A ∈ L ∞ ∩ ˙ W / , ( R , GL ( N )) such thatdiv ( A Ω − d A ) = 0 . This extends a celebrated theorem by Rivi`ere to the nonlocal case and provides conser-vation laws for a class of nonlocal equations with antisymmetric potentials, as well asstability under weak convergence.
Contents
1. Introduction 12. Preliminaries and useful tools 53. Proof of Theorem 1.1 74. Weak convergence result - Proof of Theorem 1.3 16Appendix A. Nonlocal Hodge decomposition 27Appendix B. Localization 27Appendix C. A Sobolev inequality 32Appendix D. A sequence of cut-off functions in the critical Sobolev space 33References 341.
Introduction
In the celebrated work [26] Rivi`ere showed that for two-dimensional disks D ⊂ R for anyΩ ∈ L ( D, so ( N ) ⊗ V R ), i.e., Ω ij = − Ω ji ∈ L ( D, V R ) there exists a GL ( N )-gauge, Mathematics Subject Classification.
Key words and phrases. fractional divergence, fractional div-curl lemma, fractional harmonic maps. namely a matrix-valued function
A, A − ∈ L ∞ ∩ W , ( D, GL ( N )) such thatdiv( A Ω − ∇ A ) = 0 . These are distortions of the orthonormal Uhlenbeck’s Coulomb gauges, [33], namely P ∈ L ∞ ∩ W , ( D, SO ( N )) which satisfydiv( P Ω P t − P ∇ P ) = 0 . As Rivi`ere showed in [26], the GL ( N )-gauges have the advantage that they can transformequations of the form(1.1) − ∆ u = Ω · ∇ u into a conservation law div( A ∇ u ) = div(( ∇ A − A Ω) u ) . This is important since (1.1) is the structure of the equation for harmonic maps, H -surfaces,and more generally the Euler-Lagrange equations of a large class of conformally invariantvariational functionals. The GL ( N )-gauge transform allows for regularity theory and thestudy of weak convergence [26], it also is an important tool for energy quantization, see[16].In recent years a theory of fractional harmonic maps has developed, beginning with thework by Rivi`ere and the first named author, [10, 9]. A bubbling analysis was initiated in [6].Fractional harmonic maps have a variety of applications: they appear as free boundaryof minimal surfaces or harmonic maps [24, 21, 30, 8], they are also related to nonlocalminimal surfaces [22] and to knot energies [2, 3].In [17] the second and the third named authors introduced a new approach to fractionalharmonic maps. It begins with the definition of “nonlocal one forms”. F ∈ L p ( V od R n ) if F : R n × R n → R and Z R n Z R n | F ( x, y ) | p d x d y | x − y | n < ∞ . The s -differential, which takes function u : R n → R into 1-forms, is then given by d s u ( x, y ) := u ( x ) − u ( y ) | x − y | s . The scalar product for two 1-forms, F ∈ L p ( V od R n ) and G ∈ L p ′ ( V od R n ) is then given by F · G ( x ) = Z R n F ( x, y ) G ( x, y ) dy | x − y | n . The fractional divergence div s , which takes 1-forms into functions, is then the formaladjoint to d s , namely div s F [ ϕ ] := Z R n F · d s ϕ ∀ ϕ ∈ C ∞ c ( R n ) . FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 3
For more details we refer to Section 2. With this notation in mind we now considerequations of the form(1.2) div ( d u ) = Ω · d u in R , or in index form div ( d u i ) = N X j =1 Ω ij · d u j in R , i = 1 , . . . , N , where u ∈ ( L + L ∞ ) ∩ ˙ W , ( R , R N ) and Ω ij = − Ω ji ∈ L ( V od R ).The main observation in [17] is that the above notation and the above equation are notmerely some random definitions of only analytical interest. Rather it was shown thatthe role of (1.2) for fractional harmonic maps is similar to the role of (1.1) for harmonicmaps. In [17] it was shown that there exists a div − curl Lemma in the spirit of [5], thatfractional harmonic maps into spheres satisfy a conservation law in the spirit of [15], andthat fractional harmonic maps into spheres essentially satisfy equations of the form (1.2), inthe spirit of [26], and that an analogue of Uhlenbeck’s gauge exist. In [20] this argument wasfurther pushed to equations of stationary harmonic map in higher dimensional domains.We mention that in [7] the authors found quasi conservation laws for nonlocal Schr¨odingertype systems of the form ( − ∆) / v = Ω v + f with v ∈ L ( R , R N ), Ω ∈ L ( R , so ( N )).In this article we develop the theory analogous to Rivi`ere’s GL ( N )-gauge. Our mainmotivation is that we hope this technique to be as useful for the still open question ofenergy quantization as it was in the local case in [16], a question we will study in a futurework.Applying a gauge A ∈ L ∞ ∩ ˙ W , to the equation (1.2) we find (see Lemma 4.1),div ( A ik d u k ) = (cid:16) A iℓ Ω ℓk − d A ik (cid:17) · d u k . Our main result is then the existence of the nonlocal analogue of Rivi`ere’s GL ( N )-Coulombgauge [26], namely we have Theorem 1.1.
There exists a number < σ ≪ such that the following holds.If Ω ∈ L ( V od R ) is antisymmetric, i.e., Ω ij = − Ω ji and satisfies k Ω k L ( V od R ) < σ, then there exists an invertible matrix valued function A ∈ L ∞ ∩ ˙ W , ( R , GL ( N )) such thatfor Ω A := A Ω − d A we have div (cid:0) Ω A (cid:1) = 0 . Moreover we have (1.3) [ A ] W , ( R ) - k Ω k L ( V od R ) , k A k L ∞ ( R ) - k Ω k L ( V od R ) . FRANCESCA DA LIO, KATARZYNA MAZOWIECKA, ARMIN SCHIKORRA, AND LIFENG WANG
As an immediate corollary we obtain
Corollary 1.2 (Conservation law) . Assume div ( d u k ) = Ω · d u + f, in D ′ ( R ) and Ω satisfies the condition of Theorem 1.1. Then there exists a matrix A such that for Ω A := A Ω − d A we have div (cid:16) Ad u − (Ω A ) ∗ u (cid:17) = Af, in D ′ ( R ) , where Ω ∗ A ( x, y ) := − Ω A ( y, x ) . Theorem 1.1 is applicable to the half-harmonic map system as derived [17, Proposition4.2], because of a localization result, see Proposition B.1.With the methods of Theorem 1.1 we obtain the analogue of [26, Theorem I.5], our secondmain result.
Theorem 1.3.
Assume Ω ℓ ∈ L ( V od R ) is a sequence of antisymmetric vector fields, i.e., (Ω ij ) ℓ = − (Ω ji ) ℓ , weakly convergent in L to an Ω ∈ L ( V od R ) . Assume further that f ℓ ∈ W − , ( R , R N ) converges strongly to f in W − , , and assume that u ℓ ∈ ( L + L ∞ ( R )) ∩ ˙ W , ( R , R N ) is a sequence of solutions to (1.4) ( − ∆) u ℓ = Ω ℓ · d u ℓ + f ℓ in D ′ ( R ) such that sup ℓ (cid:16) k u ℓ k L + L ∞ ( R ) + [ u ℓ ] W , ( R ) (cid:17) < ∞ . Then, up to taking a subsequence u ℓ converges weakly in ˙ W , ( R , R N ) to some u ∈ ˙ W , ( R , R N ) ∩ (( L + L ∞ )( R , R N )) , whichis a solution to ( − ∆) u = Ω · d u + f in D ′ ( R ) . Here, as usual, we denote k f k L + L ∞ ( R ) = inf f ∈ L ( R ) (cid:0) k f k L ( R ) + k f − f k L ∞ ( R ) (cid:1) . Theorem 1.3 will be proven in Section 4.
Acknowledgment.
Funding is acknowledged as follows • (FDL) Swiss National Fund, SNF200020 192062: Variational Analysis in Geometry; • (KM) FSR Incoming Post-doctoral Fellowship; • (AS) Simons Foundation (579261). FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 5 Preliminaries and useful tools
We follow the notation of [17] for the nonlocal operators. For readers convenience we recallit here. We write M ( V od R n ) for the space of vector fields F : R n × R n → R measurablewith respect to the d x d y | x − y | n measure, where “ od ” stands for “off diagonal”.For two vector fields F, G ∈ M ( V od R n ) the scalar product is defined as F · G ( x ) := Z R n F ( x, y ) G ( x, y ) d y | x − y | n . For any p > L p -space on vector fields F : R n × R n → R is induced by thenorm k F k L p ( V od R n ) := (cid:18)Z R n Z R n | F ( x, y ) | p d x d y | x − y | n (cid:19) p and for D ⊂ R n we define k F k L p ( V od D ) := (cid:18)Z Z ( D × R n ) ∪ ( R n × D ) | F ( x, y ) | p d x d y | x − y | n (cid:19) p . For f : R n → R we let the s -gradient d s : M ( R n ) → M ( V od R n ) to be d s f ( x, y ) := f ( x ) − f ( y ) | x − y | s . Observe that with this notation we have k d s f k L p ( V od R n ) = [ f ] W s,p ( R n ) , where [ f ] W s,p ( R n ) = (cid:18)Z R n Z R n | f ( x ) − f ( y ) | p | x − y | n + sp d x d y (cid:19) /p is the Gagliardo–Slobodeckij seminorm.We define the fractional s -divergence in the distributional way. We let for F ∈ M ( V od R n )div s F [ ϕ ] := Z R n Z R n F ( x, y ) d s ϕ ( x, y ) d x d y | x − y | n , ϕ ∈ C ∞ c ( R n ) . With this notation we have div s d s = − (∆) s , i.e., Z R n d s f · d s g ( x ) d x = Z R ( − ∆) s f ( x ) g ( x ) d x, where the fractional Laplacian is defined as( − ∆) s f ( x ) := 2 P.V. Z R n f ( x ) − f ( y ) | x − y | s d y | x − y | n . A simple observation is the following
FRANCESCA DA LIO, KATARZYNA MAZOWIECKA, ARMIN SCHIKORRA, AND LIFENG WANG
Lemma 2.1.
Let F ∈ D ′ ( V od R n ) then we define F ∗ ( x, y ) := − F ( y, x ) . If div s F = 0 then div s F ∗ = 0 .Moreover, for any F ∈ D ′ ( V od R n ) and u ∈ W s, ( R n ) we have div s ( F u ) = div s ( F ) u + F ∗ · d s u. Proof.
We have F ( x, y ) u ( x )( ϕ ( x ) − ϕ ( y )) = F ( x, y )( u ( x ) ϕ ( x ) − u ( y ) ϕ ( y )) − F ( x, y )( u ( x ) − u ( y )) ϕ ( y ) . Thus, Z R n Z R n F ( x, y ) u ( x )( ϕ ( x ) − ϕ ( y )) | x − y | n + s d y d x = Z R n Z R n F ( x, y )( u ( x ) ϕ ( x ) − u ( y ) ϕ ( y )) | x − y | n + s d y d x − Z R n Z R n F ( x, y )( u ( x ) − u ( y )) ϕ ( y ) | x − y | n + s d y d x. As for the latter term we have − Z R n Z R n F ( x, y )( u ( x ) − u ( y )) ϕ ( y ) | x − y | n + s d y d x = Z R n Z R n − F ( y, x )( u ( x ) − u ( y )) ϕ ( x ) | x − y | n + s d y d x = Z R n Z R n F ∗ ( x, y )( u ( x ) − u ( y )) ϕ ( x ) | x − y | n + s d y d x. (cid:3) We also denote | D s,q f | ( x ) := (cid:18)Z R n | f ( x ) − f ( y ) | q | x − y | n + sq d y (cid:19) q . We will be using the following “Sobolev embedding” theorem.
Theorem 2.2.
Let s ∈ (0 , , t ∈ ( s, , and let p, p ∗ > satisfy s − np ∗ = t − np , where q > with p ∗ > nqn + sq . Then we have (2.1) k|D s,q f |k L p ∗ ( R n ) - k ( − ∆) t f k L p ( R n ) and for any r ∈ [1 , ∞ ](2.2) k|D s,q f |k L ( p ∗ ,r ) ( R n ) - k ( − ∆) t f k L ( p,r ) ( R n ) . FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 7
For the proof see Appendix C.We will also need the following Wente’s inequality from [17].
Lemma 2.3 ([17, Corollary 2.3]) . Let s ∈ (0 , , p > , and let p ′ be the H¨older conjugateof p . Assume moreover that F ′ ∈ L p ( V od R ) and g ∈ W s,p ′ ( R ) with div s F = 0 . Let R bea linear operator such that for some Λ > satisfies | R [ ϕ ] | ≤ Λ k ( − ∆) ϕ k L (2 , ∞ ) ( R ) , where L (2 , ∞ ) ( R ) denote the weak L space. Then any distributional solution u ∈ ˙ W , ( R ) to ( − ∆) u = F · d s g + R in R is continuous. Moreover if lim x →±∞ | u ( x ) | = 0 , then we have the estimate (2.3) k u k L ∞ ( R ) + k d u k L ( V od R ) - k F k L p ( V od R ) k d s g k L p ′ ( V od R ) + Λ . Our proof will also be based on the following choice of a good gauge.
Theorem 2.4 ([17, Theorem 4.4]) . For Ω ij = − Ω ji ∈ L ( V od R ) there exists P ∈ ˙ W ( R , SO ( N )) such that div Ω Pij = 0 for all i, j ∈ { , . . . , N } , where Ω P = 12 (cid:16) d P ( x, y ) (cid:0) P T ( y ) + P T ( x ) (cid:1) − P ( x )Ω( x, y ) P T ( y ) − P ( y )Ω( x, y ) P T ( x ) (cid:17) and (2.4) [ P ] W , ( R ) - k Ω k L ( V od R ) . Proof of Theorem 1.1
In this section we prove Theorem 1.1. We will be looking for an A in the form A = ( I + ε ) P ,where P is chosen to be the good gauge from Theorem 2.4. The idea to take perturbationof rotations of the form ( I + ε ) P has been taken from [27] in the context of local Schr¨odingerequations with antisymmetric potentials. This has been also exploited in [7]. Lemma 3.1.
Assume that A = ( I + ε ) P .Then for Ω P ( x, y ) = 12 (cid:16) d P ( x, y ) (cid:0) P T ( y ) + P T ( x ) (cid:1) − P ( x )Ω( x, y ) P T ( y ) − P ( y )Ω( x, y ) P T ( x ) (cid:17) . we have A ( x )Ω( x, y ) − d A ( x, y ) = − ( I + ε ( x )) Ω P ( x, y ) P ( y ) − d ε ( x, y ) P ( y ) + R ε ( x, y ) , FRANCESCA DA LIO, KATARZYNA MAZOWIECKA, ARMIN SCHIKORRA, AND LIFENG WANG where R ε is given by the formula R ε ( x, y ) := 12 ( I + ε ( x )) (cid:18) d P ( x, y ) d P T ( x, y ) − P ( x ) Ω( x, y ) (cid:0) P T ( x ) − P T ( y ) (cid:1) + ( P ( x ) − P ( y )) Ω( x, y ) P T ( x ) (cid:19) P ( y ) . (3.1) Proof.
Recall that d ( f g )( x, y ) = d f ( x, y ) g ( y ) + f ( x ) d g ( x, y ) . Thus, applying this to d (( I + ε ) P )( x, y ) we get A ( x )Ω( x, y ) − d A ( x, y )= ( I + ε ( x )) P ( x )Ω( x, y ) − d (( I + ε ) P ) ( x, y )= ( I + ε ( x )) (cid:16) P ( x ) Ω( x, y ) − d P ( x, y ) (cid:17) − d ε ( x, y ) P ( y )= − ( I + ε ( x )) (cid:16) d P ( x, y ) P T ( y ) − P ( x ) Ω( x, y ) P T ( y ) (cid:17) P ( y ) − d ε ( x, y ) P ( y ) . (3.2)Next we observe that d P ( x, y ) P T ( y ) − P ( x ) Ω( x, y ) P T ( y )= 12 (cid:16) d P ( x, y ) (cid:0) P T ( x ) + P T ( y ) (cid:1) − P ( x ) Ω( x, y ) P T ( y ) − P ( y ) Ω( x, y ) P T ( x ) (cid:17) − (cid:16) d P ( x, y ) (cid:0) P T ( x ) − P T ( y ) (cid:1) − P ( x ) Ω( x, y ) (cid:0) P T ( x ) − P T ( y ) (cid:1) + ( P ( x ) − P ( y )) Ω( x, y ) P T ( x ) (cid:17) . (3.3)That is, plugging in (3.3) into (3.2) we get the claim for R ε ( x, y ) := 12 ( I + ε ( x )) (cid:18) d P ( x, y ) d P T ( x, y ) − P ( x ) Ω( x, y ) (cid:0) P T ( x ) − P T ( y ) (cid:1) + ( P ( x ) − P ( y )) Ω( x, y ) P T ( x ) (cid:19) P ( y ) . (cid:3) Lemma 3.2.
Assume that we have ε ∈ L ∞ ∩ ˙ W / , ( R ) , a ∈ ˙ W / , ( R ) , and B ∈ L ( V od R ) satisfying the equations (3.4) − ( I + ε ( x )) Ω P ( x, y ) P ( y ) − d ε ( x, y ) P ( y ) + R ε ( x, y ) = d a ( x, y ) + B ( x, y ) FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 9 and − div (cid:0) ( I + ε ( x )) Ω P ( x, y ) (cid:1) − div (cid:16) d ε ( x, y ) (cid:17) + div ( R ε ( x, y ) P T ( y ))= div (cid:0) B ( x, y ) P T ( y ) (cid:1) , (3.5) with (3.6) [ P ] W / , ( R ) < σ. Then, for sufficiently small σ we have a = const .Proof. We multiply (3.4) by P T ( y ) from the right and take the -divergence on both sides,then subtracting (3.5) we obtain(3.7) div ( d a ( x, y ) P T ( y )) = 0 . We use nonlocal Hodge decompostion Lemma A.1 and get the existence of functions ˜ a ∈ ˙ W , ( R ), ˜ B ∈ L ( V od R ), such that lim | x |→∞ ˜ a ( x ) = 0(3.8) d a ( x, y ) P T ( y ) = d ˜ a ( x, y ) + ˜ B ( x, y ) , and (recall | P | = 1)(3.9) div ˜ B = 0 and k ˜ B k L ( V od R ) - k d a k L ( V od R ) . Thus, taking the -divergence in (3.8) we obtain0 = div ( d a ( x, y ) P T ( y )) = div ( d ˜ a ( x, y ) + ˜ B ( x, y )) = div ( d ˜ a ) = ( − ∆) ˜ a. This gives, ( − ∆) ˜ a = 0, recall that lim | x |→∞ ˜ a ( x ) = 0, thus combined with the Liouvilletheorem for fractional Laplacian [11, Theorem 1.1], we obtain ˜ a = 0. Thus (3.8) becomes d a ( x, y ) P T ( y ) = ˜ B ( x, y ) . That is d a ( x, y ) = ˜ B ( x, y ) P ( y ) . Taking the -divergence we obtain by Lemma 2.1(3.10) ( − ∆) a = ˜ B ∗ · d P, since on the righ-hand side we have a div-curl term we can apply fractional Wente’s in-equality, Lemma 2.3, and obtain from (2.3) k d a k L ( V od R ) - k ˜ B k L ( V od R ) k d P k L ( V od R ) . Combining this with (3.9) and (3.6) we get k d a k L ( V od R ) - σ k d a k L ( V od R ) , which implies for sufficiently small σ that k d a k L ( V od R ) = [ a ] W / , ( R ) = 0 and thus a ≡ const . (cid:3) Now we will focus on showing that there exists a solution to the equations (3.4) and (3.5).We will do this by using the Banach fixed point theorem.
Proposition 3.3.
Let Ω ∈ L ( V od R ) be anitsymmetric. There is a number < σ ≪ such that the following holds:Take P ∈ ˙ W , ( R , SO ( N )) and Ω P ∈ L ( V od R ) from Theorem 2.4. Let us assume that (3.11) [ P ] W / , ( R ) + k Ω k L ( V od R ) < σ. Then, there exist ε ∈ L ∞ ∩ ˙ W / , ( R ) , a ∈ ˙ W / , ( R ) , and B ∈ L ( V od R ) that solve theequations (3.12) ( − ( I + ε ( x )) Ω P ( x, y ) P ( y ) + d ε ( x, y ) P ( y ) + R ε ( x, y ) = d a ( x, y ) + B ( x, y ) − div (cid:0) ( I + ε ( x ))Ω P ( x, y ) (cid:1) + div ( d ε ( x, y )) + div ( R ε ( x, y ) P T ( y )) = div (cid:0) BP T ( y ) (cid:1) , where R ε is defined in (3.1) .Moreover, ε satisfies the estimate (3.13) k ε k L ∞ ( R ) + [ ε ] W , ( R ) - k Ω k L ( V od R ) . We will need the following remainder terms estimates.
Lemma 3.4.
We have the following estimates (cid:12)(cid:12)(cid:12) div ( R ε P T )[ ϕ ] (cid:12)(cid:12)(cid:12) - (1 + k ε k L ∞ ( R ) )( k Ω k L ( V od R ) + [ P ] W / , ( R ) )[ P ] W / , ( R ) k ( − ∆) ϕ k L (2 , ∞ ) ( R ) (3.14) and | div ( R ε − R ε ) P T )[ ϕ ] | - k ε − ε k L ∞ ( R ) ( k Ω k L ( V od R ) + [ P ] W / , ( R ) )[ P ] W / , ( R ) k ( − ∆) ϕ k L (2 , ∞ ) ( R ) . (3.15) FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 11
Proof.
We observe that for any ϕ ∈ C ∞ c ( R ) we have (cid:12)(cid:12)(cid:12) div ( R ε P T )[ ϕ ] (cid:12)(cid:12)(cid:12) - (cid:12)(cid:12)(cid:12)(cid:12)Z R Z R ( I + ε ( x )) (cid:16) d P ( x, y ) d P T ( x, y ) (cid:17) d ϕ ( x, y ) d x d y | x − y | (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R Z R ( I + ε ( x )) (cid:18) P ( x ) Ω( x, y ) (cid:0) P T ( x ) − P T ( y ) (cid:1) d ϕ ( x, y ) d x d y | x − y | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R Z R ( I + ε ( x )) ( P ( x ) − P ( y )) Ω( x, y ) P T ( x ) d ϕ ( x, y ) d x d y | x − y | (cid:12)(cid:12)(cid:12)(cid:12) - (1 + k ε k L ∞ ) Z R Z R (cid:16) | d P ( x, y ) | | d ϕ ( x, y ) | + | Ω( x, y ) | | d P ( x, y ) | | d ϕ ( x, y ) | (cid:17) d x d y | x − y | = (1 + k ε k L ∞ ) ( I + II ) . (3.16)Let M be the Hardy–Littlewood maximal function and let α ∈ (0 , | f ( x ) − f ( y ) | - | x − y | α (cid:0) M (( − ∆) α f )( x ) + M (( − ∆) α f )( y ) (cid:1) of the well known inequality, see [4, 14] | f ( x ) − f ( y ) | - | x − y | ( M|∇ f | ( x ) + M|∇ f | ( y )) . We begin with the estimate of the first term on the right-hand side of (3.16).We observe that by (3.17) and by the symmetry of the integrals we obtain I := Z R Z R | d P ( x, y ) | | d ϕ ( x, y ) | d x d y | x − y | - Z R |M (( − ∆) ϕ )( x ) | Z R | d P ( x, y ) | d x d y | x − y | . (3.18)Applying H¨older’s inequality (for Lorentz spaces) we obtain Z R |M (( − ∆) ϕ )( x ) | Z R | d P ( x, y ) | d x d y | x − y | - k ( − ∆) ϕ k L (2 , ∞ ) k|D , P | k L (2 , = k ( − ∆) ϕ k L (2 , ∞ ) k|D , P |k L (4 , , (3.19)where we used the notation from Section 2: for s ∈ (0 ,
1) and q > |D s,q f | ( x ) := (cid:18)Z R | f ( x ) − f ( y ) | q | x − y | sq d y (cid:19) q . Applying Theorem 2.2, (2.2) for t = we get(3.20) k|D , P |k L (4 , - k ( − ∆) P k L (2 , - k ( − ∆) P k L = [ P ] W / , . Thus, combining (3.18), (3.19), and (3.20) we obtain(3.21) I = Z R Z R | d P ( x, y ) | | d ϕ ( x, y ) | d x d y | x − y | - [ P ] W / , ( R ) k ( − ∆) ϕ k L (2 , ∞ ) ( R ) . As for the second term of (3.16) we have II := Z R Z R | Ω( x, y ) || d P ( x, y ) || d ϕ ( x, y ) | d x d y | x − y | - k Ω k L ( V od R ) (cid:18)Z R Z R | d P ( x, y ) | | d ϕ ( x, y ) | d x d y | x − y | (cid:19) . (3.22)Applying once again (3.17) we obtain Z R Z R | d P ( x, y ) | | d ϕ ( x, y ) | d x d y | x − y | - Z R Z R (cid:16) M (( − ∆) ϕ )( x ) + M (( − ∆) ϕ )( y ) (cid:17) | d P ( x, y ) | d x d y | x − y | - Z R (cid:16) M (( − ∆) ϕ )( x ) (cid:17) Z R | d P ( x, y ) | d x d y | x − y | . (3.23)Using H¨older’s inequality and then Sobolev embedding we get Z R (cid:16) M (( − ∆) ϕ )( x ) (cid:17) Z R | d P ( x, y ) | d x d y | x − y | - k ( M ( − ∆) ϕ ) k L (2 , ∞ ) ( R ) k|D , P | k L (2 , ( R ) - k ( − ∆) ϕ k L (4 , ∞ ) ( R ) k|D , P |k L (4 , ( R ) - k ( − ∆) ϕ k L (2 , ∞ ) ( R ) k| ( − ∆) P k L (2 , ( R ) , (3.24)where for the estimate of the last term we used again Theorem 2.2, (2.2), with t = .Combining (3.22), (3.23), and (3.24) we obtain II = Z R Z R | Ω( x, y ) || d P ( x, y ) || d ϕ ( x, y ) | d x d y | x − y | - k Ω k L ( V od R ) k ( − ∆) ϕ k L (2 , ∞ ) ( R ) [ P ] W / , ( R ) . (3.25)Finally, from (3.16), (3.21), and (3.25) we get (cid:12)(cid:12)(cid:12) div ( R ε P T )[ ϕ ] (cid:12)(cid:12)(cid:12) - (1 + k ε k L ∞ ( R ) ) (cid:16) k Ω k L ( V od R ) + [ P ] W / , ( R ) (cid:17) [ P ] W / , ( R ) k ( − ∆) ϕ k L (2 , ∞ ) ( R ) . This finishes the proof of (3.14).In order to prove (3.15) we observe (cid:12)(cid:12)(cid:12) div ( R ε − R ε ) P T )[ ϕ ] (cid:12)(cid:12)(cid:12) - k ε − ε k L ∞ ( I + II ) . Thus, in order to conclude it suffices to apply the estimates (3.21) and (3.25). (cid:3)
FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 13
Proof of Proposition 3.3.
Let X = L ∞ ∩ ˙ W , ( R ).For any ε ∈ X we have A = (1 + ε ) P ∈ L ∞ ∩ ˙ W ( R ), which implies A Ω − d A ∈ L ( V od R )and thus, from Lemma 3.1, we have − (cid:0) ( I + ε ( x, y )) Ω P ( x, y ) P ( y ) (cid:1) + (cid:16) d ε ( x, y ) P ( y ) (cid:17) + R ε ( x, y ) ∈ L ( ^ od R ) . We apply for this term the nonlocal Hodge decomposition, Lemma A.1: given ε ∈ X wefind a ( ε ) ∈ W , ( R ) and B ( ε ) ∈ L ( V od R ) with div B ( ε ) = 0 satisfying − (cid:0) ( I + ε ( x, y )) Ω P ( x, y ) P ( y ) (cid:1) + (cid:16) d ε ( x, y ) P ( y ) (cid:17) + R ε ( x, y )= d a ( ε )( x, y ) + B ( ε )( x, y )(3.26)with the estimates k B ( ε ) k L ( V od R ) + [ a ( ε )] W , ( R ) - (1 + k ε k L ∞ ( R ) )([ P ] W / , ( R ) + k Ω k L ( V od R ) ) + [ ε ] W / , ( R ) . (3.27)Similarly, if for any two ε , ε ∈ X we consider the difference of the corresponding equations(3.26) we get k B ( ε ) − B ( ε ) k L ( V od R ) - k ε − ε k L ∞ ( R ) ([ P ] W / , ( R ) + k Ω k L ( V od R ) ) + [ ε − ε ] W / , ( R ) . (3.28)Now we define the mapping T : X → X as the solution to − div (cid:0) ( I + ε ( x )) Ω P ( x, y ) (cid:1) + div (cid:16) d T ( ε )( x, y ) (cid:17) + div ( R ε ( x, y ) P T ( y ))= div (cid:0) B ( ε )( x, y ) P T ( y ) (cid:1) (3.29)with lim | x |→∞ T ( ε )( x ) = 0.(3.29) can be rewritten as( − ∆) T ( ε ) = div (cid:0) B ( ε ) P T (cid:1) + div (cid:0) ( I + ε ) Ω P (cid:1) − div ( R ε P T )= ( B ( ε )) ∗ · d P T + d ( I + ε ) · (Ω P ) ∗ − div ( R ε P T ) , (3.30)we used in the second inequality Lemma 2.1.We observe that on the right-hand we have fractional div - curl -terms: div ( B ( ε )) ∗ = 0 anddiv (Ω P ) ∗ = 0. Let us denoteΛ ε := (1 + k ε k L ∞ ( R ) )( k Ω k L ( R ) + [ P ] W / , ( R ) )[ P ] W / , ( R ) . By Lemma 3.4, (3.14), the rest term in (3.30) satisfies (cid:12)(cid:12)(cid:12) div ( R ε P T )[ ϕ ] (cid:12)(cid:12)(cid:12) - Λ ε k ( − ∆) ϕ k L (2 , ∞ ) ( R ) . Thus, we may apply the nonlocal Wente’s lemma, i.e., Lemma 2.3 and obtain k T ( ε ) k L ∞ ( R ) + [ T ( ε )] W / , ( R ) - k ( B ( ε )) ∗ k L ( V od R ) [ P ] W / , ( R ) + [ ε ] W / , ( R ) k (Ω P ) ∗ k L ( V od R ) + Λ ε = k B ( ε ) k L ( V od R ) [ P ] W / , ( R ) + [ ε ] W / , ( R ) k Ω P k L ( V od R ) + Λ ε . (3.31)Moreover, let ε , ε ∈ X , then we have( − ∆) ( T ( ε ) − T ( ε ))= div (cid:0) ( B ( ε ) − B ( ε )) P T ( y ) (cid:1) + div (cid:0) ( ε − ε ) Ω P (cid:1) − div (( R ε − R ε ) P T )= (( B ( ε )) ∗ − ( B ( ε )) ∗ ) · d P T + d ( ε − ε ) · (Ω P ) ∗ − div (( R ε − R ε ) P T ) , (3.32)where we have used again Lemma 2.1.Again, we observe thatdiv (( B ( ε )) ∗ − ( B ( ε )) ∗ ) = 0 and div (Ω P ) ∗ = 0 , and from Lemma 3.4, (3.15), we may estimate the rest term in (3.32)(3.33) | div ( R ε − R ε ) P T )[ ϕ ] | - Λ ε ,ε k ( − ∆) ϕ k L (2 , ∞ ) ( R ) , where(3.34) Λ ε ,ε := k ε − ε k L ∞ ( R ) ([ P ] W / , ( R ) + k Ω k L ( R ) )[ P ] W / , ( R ) . Therefore we may apply the nonlocal Wente’s Lemma 2.3 for equation (3.32) and obtain k T ( ε ) − T ( ε ) k L ∞ ( R ) + [ T ( ε ) − T ( ε )] W / , ( R ) - k B ( ε ) − B ( ε ) k L ( V od R ) [ P ] W / , ( R ) + [ ε − ε ] W / , ( R ) k Ω p k L ( V od R ) + Λ ε ,ε . (3.35)Combining (3.35) with (3.28) and (3.34) we get k T ( ε ) − T ( ε ) k L ∞ ( R ) + [ T ( ε ) − T ( ε )] W / , ( R ) - k ε − ε k L ∞ ( R ) (cid:16) [ P ] W / , ( R + k Ω k L ( V od R ) (cid:17) [ P ] W / , ( R ) + [ ε − ε ] W / , ( R ) (cid:16) [ P ] W / , ( R + k Ω k L ( V od R ) (cid:17) - ( k ε − ε k L ∞ ( R ) + [ ε − ε ] W / , ( R ) | ) σ, where in the last inequality we used (3.11).Thus, taking σ small enough we obtain k T ( ε ) − T ( ε ) k L ∞ ( R ) + [ T ( ε ) − T ( ε )] W / , ( R ) ≤ λ (cid:0) k ε − ε k L ∞ ( R ) + [ ε − ε ] W / , ( R ) (cid:1) , for a 0 < λ <
1, which implies that T is a contraction. Consequently, by Banach fixedpoint theorem, there exists a unique ε ∈ X , such that T ( ε ) = ε . That is we have a solution FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 15 T ( ε ) = ε , which is a solution to − (cid:0) ( I + ε ) Ω P P (cid:1) + (cid:16) d εP (cid:17) + R ε = d a ( ε ) + B ( ε ) − div (cid:0) ( I + ε ) Ω P (cid:1) + div (cid:16) d ε (cid:17) + div ( R ε P T ) = div (cid:0) B ( ε ) P T (cid:1) . Moreover, combining (3.31) with (3.27) and (3.11) we obtain the following estimate on ε (3.36) k ε k L ∞ ( R ) + [ ε ] W , ( R ) - σ k ε k L ∞ ( R ) + σ [ ε ] W , ( R ) + k Ω k L ( V od R ) + [ P ] W , ( R ) , which gives for sufficiently small σ k ε k L ∞ ( R ) + [ ε ] W , ( R ) - k Ω k L ( V od R ) + [ P ] W , ( R ) . (cid:3) Proof of Theorem 1.1.
By Proposition 3.3 we obtain the existence of an ε ∈ L ∞ ∩ ˙ W , ( R ), a ∈ ˙ W , ( R ), B ∈ L ( V od R ) with div B = 0 satisfying the equations solution T ( ε ) = ε ,which is a solution to − (cid:0) ( I + ε ) Ω P P (cid:1) + (cid:16) d εP (cid:17) + R ε = d a + B − div (cid:0) ( I + ε ) Ω P (cid:1) + div (cid:16) d ε (cid:17) + div ( R ε P T ) = div (cid:0) BP T (cid:1) , where P ∈ ˙ W , ( R , SO ( N )) and Ω P ∈ L ( V od R ) are taken from Theorem 2.4 and[ P ] W / , ( R ) - k Ω k L ( V od R ) ≤ σ .By Lemma 3.2 we have for sufficiently small σ − (cid:0) ( I + ε ) Ω P P (cid:1) + (cid:16) d εP (cid:17) + R ε = B. Thus, defining for ε from Proposition 3.3, A := ( I + ε ) P , we have by Lemma 3.1 A Ω − d A = B. The invertibility of A follows from the invertibility of P and I + ε . Finally, since A =( I + ε ) P , we obtain from (3.13) and (2.4) the estimates[ A ] W , ( R ) - (1 + k ε k L ∞ )[ P ] W , ( R ) + [ ε ] W , ( R ) - k Ω k L ( V od R ) , and k A k L ∞ ( R ) - k Ω k L ( V od R ) . This finishes the proof. (cid:3) Weak convergence result - Proof of Theorem 1.3
Lemma 4.1.
Assume Ω ∈ L ( V od R ) . u ∈ ˙ W , ( R , R N ) ∩ ( L + L ∞ ( R )) is a solution to (4.1) ( − ∆) u i = Ω · d u if and only if for any invertible matrix valued function A, A − ∈ L ∞ ∩ ˙ W , ( R , GL ( N )) , div ( A ik d u k ) = (cid:16) A ij Ω jk − d A ik (cid:17) · d u k . Proof.
We give only proof in one direction, the other direction follows by testing with A − ψ .Assume u is a solution to (4.1), then Z R Z R A ik ( x ) (cid:0) u k ( x ) − u k ( y ) (cid:1) ( ϕ ( x ) − ϕ ( y )) d x d y | x − y | = Z R Z R (cid:0) u k ( x ) − u k ( y ) (cid:1) ( A ik ( x ) ϕ ( x ) − A ik ( y ) ϕ ( y )) d x d y | x − y | + Z R Z R (cid:0) u k ( x ) − u k ( y ) (cid:1) ( A ik ( y ) − A ik ( x )) ϕ ( y ) d x d y | x − y | . By the equation (4.1) Z R Z R (cid:0) u k ( x ) − u k ( y ) (cid:1) ( A ik ( x ) ϕ ( x ) − A ik ( y ) ϕ ( y )) d x d y | x − y | = Z R Z R A ik ( x ) Ω kj ( x, y ) d u j ( x, y ) ϕ ( x ) d x d y | x − y | . Moreover, by symmetry, Z R Z R (cid:0) u k ( x ) − u k ( y ) (cid:1) ( A ik ( y ) − A ik ( x )) ϕ ( y ) d x d y | x − y | = Z R Z R (cid:0) u k ( x ) − u k ( y ) (cid:1) ( A ik ( y ) − A ik ( x )) ϕ ( x ) d x d y | x − y | = − Z R Z R d A ik ( x, y ) d u k ( x, y ) ϕ ( x ) d x d y | x − y | . Thus, we have Z R Z R A ik ( x ) (cid:0) u k ( x ) − u k ( y ) (cid:1) ( ϕ ( x ) − ϕ ( y )) d x d y | x − y | = Z R Z R (cid:16) A ij ( x ) Ω jk ( x, y ) − d A ik ( x, y ) (cid:17) d u k ( x, y ) ϕ ( x ) d x d y | x − y | . (cid:3) In a first step we prove the “local version” of Theorem 1.3.
FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 17
Proposition 4.2.
Let σ > be the number from Theorem 1.1. Let { u ℓ } ℓ ∈ N be a sequenceas in Theorem 1.3 of solutions to ( − ∆) u ℓ = Ω ℓ · d u ℓ + f ℓ in D ′ ( R ) . Additionally let us assume that for some bounded interval D ⊂ R we have (4.2) sup ℓ k Ω ℓ k L ( V od D ) < σ. Then ( − ∆) u = Ω · d u + f in D ′ ( D ) . Proof.
Let us define Ω
D,ℓ := χ D ( x ) χ D ( y )Ω ℓ ∈ L ( V od R ). Then by (4.2) we have(4.3) k Ω D,ℓ k L ( V od R ) ≤ k Ω ℓ k L ( V od D ) < σ. By Theorem 1.1 for Ω
D,ℓ there exists a gauge A ℓ such that(4.4) div (Ω A ℓ D,ℓ ) = 0 , where Ω A ℓ D,ℓ := A ℓ Ω D,ℓ − d A ℓ .Let D ⊂⊂ D be an open set.By assumption and Lemma 4.1 we have for any ψ ∈ C ∞ c ( D ) Z R A ℓ d u ℓ · d ψ = Z R Ω A ℓ ℓ d u ℓ ψ + f ℓ [ A ℓ ψ ] . Let us denote Ω D c ,ℓ := Ω ℓ − Ω D,ℓ . Then we have Z R A ℓ d u ℓ · d ψ = Z R Ω A ℓ D,ℓ · d u ℓ ψ + Z R A ℓ Ω D c ,ℓ · d u ℓ ψ + f ℓ [ A ℓ ψ ] . By Lemma 2.1, we have div (Ω A ℓ D,ℓ u ℓ ) = (cid:16) Ω A ℓ D,ℓ (cid:17) ∗ · d u ℓ with div (cid:16)(cid:16) Ω A ℓ D,ℓ (cid:17) ∗ (cid:17) = 0, thus(4.5) Z R A ℓ d u ℓ · d ψ = Z R (cid:16) Ω A ℓ D,ℓ (cid:17) ∗ · u ℓ d ψ + Z R A ℓ Ω D c ,ℓ · d u ℓ ψ + f ℓ [ A ℓ ψ ] . We will pass with ℓ → ∞ in (4.5). Roughly speaking, the convergence of most of the termswill be a result of a combination of weak-strong convergence. We first observe that byTheorem 1.1 we have k A ℓ k ˙ W , ( R ) - k Ω D,ℓ k L ( V od R ) ≤ σ and k A ℓ k L ∞ ( R ) - σ. Thus, sup ℓ k A ℓ k ˙ W , ( R ) < ∞ and sup ℓ k A ℓ k L ∞ ( R ) < ∞ . Up to taking a subsequence weobtain(4.6) A ℓ ⇀ A weakly in ˙ W , ( R , R N ) , A ℓ → A locally strongly in L , where we used the Rellich–Kondrachov’s compact embedding theorem and A ∈ L ∞ ∩ ˙ W , ( R , GL ( N )). By the pointwise a.e. convergence we have k A k L ∞ ( R ) - σ . By (4.3) we also have up to a subsequenceΩ
D,ℓ ⇀ Ω D weakly in L ( ^ od R ) , where Ω D ∈ L ( V od R ).By assumptions of the Theorem we also have, up to a subsequence, u ℓ ⇀ u weakly in ˙ W , ( R ) , u ℓ → u locally strongly in L , where u ∈ ˙ W , ( R , R N ). Step 1.
We claim that (up to a subsequence)(4.7) lim ℓ →∞ Z R A ℓ d u ℓ · d ψ = Z R Ad u · d ψ. Indeed, we observe Z R A ℓ d u ℓ · d ψ − Z R Ad u · d ψ = Z R ( A ℓ − A ) d u ℓ · d ψ + Z R A ( d u ℓ − d u ) · d ψ. (4.8)By weak convergence of d u ℓ in L ( V od R ) we have(4.9) lim ℓ →∞ Z R A ( d u ℓ − d u ) · d ψ = 0 . As for the first term of (4.8) we observe that for some large R ≫
1, such that in particular D ⊂ B ( R ), we have since supp ψ ⊂ D , Z R Z R ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y = Z B ( R ) Z B ( R ) ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y + Z R \ B ( R ) Z B ( R ) ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y + Z B ( R ) Z R \ B ( R ) ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y. (4.10)By strong convergence in L of A ℓ on compact domains, we havelim ℓ →∞ Z B ( R ) Z B ( R ) ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y - lim ℓ →∞ k A ℓ − A k L ( B ( R )) k ψ k Lip [ u ℓ ] W , ( B ( R )) = 0(4.11) FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 19 and lim ℓ →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z R \ B ( R ) Z B ( R ) ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y (cid:12)(cid:12)(cid:12)(cid:12) - lim ℓ →∞ k A ℓ ( x ) − A ( x ) k L ( B ( R ) k ψ k Lip [ u ℓ ] W , ( R ) = 0 . (4.12)For the last term of (4.10), we observe that if y ∈ supp ψ and x ∈ R \ B ( R ), then we have | x − y | % | x | with a constant independent of R . (cid:12)(cid:12)(cid:12)(cid:12)Z B ( R ) Z R \ B ( R ) ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y (cid:12)(cid:12)(cid:12)(cid:12) - ( k A ℓ k L ∞ + k A k L ∞ ) k ψ k L ∞ Z D Z R \ B ( R ) | u ℓ ( x ) | + | u ℓ ( y ) | | x | d x d y - ( k A ℓ k L ∞ + k A k L ∞ ) k ψ k L ∞ (cid:18) k u ℓ k L ( D ) Z R \ B ( R )
11 + | x | d x (cid:19) + ( k A ℓ k L ∞ + k A k L ∞ ) k ψ k L ∞ k u ℓ k L ∞ + L ( R ) Z R \ B ( R ) d x | x | + (cid:18)Z R \ B ( R ) d x (1 + | x | ) (cid:19) !! - ( k A ℓ k L ∞ + k A k L ∞ ) k ψ k L ∞ (cid:0) k u k L ( D ) + k u ℓ k L ∞ + L ( R ) (cid:1) R − . So we have(4.13) lim R →∞ sup ℓ (cid:12)(cid:12)(cid:12)(cid:12)Z B ( R ) Z R \ B ( R ) ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y (cid:12)(cid:12)(cid:12)(cid:12) = 0 , which finishes the Thus, by (4.10), (4.11), (4.12), and (4.13) we obtain the convergence ofthe second term of (4.8), i.e.,(4.14) lim ℓ →∞ Z R Z R ( A ℓ ( x ) − A ( x )) ( u ℓ ( x ) − u ℓ ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y = 0 . Thus, combining (4.8), (4.9), and (4.14) we obtain the claim (4.7).
Step 2.
We claim that (up to a subsequence)(4.15) lim ℓ →∞ Z R (cid:16) Ω A ℓ D,ℓ (cid:17) ∗ · u ℓ d ψ = Z R (cid:0) Ω AD (cid:1) ∗ · ud ψ, where Ω AD := A Ω D − d A . Indeed, we write Z R (cid:16) Ω A ℓ D,ℓ (cid:17) ∗ · u ℓ d ψ − Z R (cid:0) Ω AD (cid:1) ∗ · ud ψ = Z R Z R (cid:16)(cid:16) Ω A ℓ D,ℓ (cid:17) ∗ ( x, y ) u ℓ ( x ) − (cid:0) Ω AD (cid:1) ∗ ( x, y ) u ( x ) (cid:17) ψ ( x ) − ψ ( y ) | x − y | d x d y | x − y | = Z R Z R (cid:16) d A ℓ ( y, x ) u ℓ ( x ) − d A ( y, x ) u ( x ) (cid:17) ψ ( x ) − ψ ( y ) | x − y | d x d y | x − y |− Z R Z R ( A ℓ ( y )Ω D,ℓ ( y, x ) − A ( y )Ω D ( y, x )) ψ ( x ) − ψ ( y ) | x − y | d x d y | x − y | . Now, proceeding exactly as in Step 1 we obtainlim ℓ →∞ Z R Z R (cid:16) d A ℓ ( y, x ) u ℓ ( x ) − d A ( y, x ) u ( x ) (cid:17) ψ ( x ) − ψ ( y ) | x − y | d x d y | x − y | = 0and Z R Z R ( A ℓ ( y )Ω D,ℓ ( y, x ) − A ( y )Ω D ( y, x )) ψ ( x ) − ψ ( y ) | x − y | d x d y | x − y | = 0 . This, finishes the proof of (4.15).
Step 3.
We claim that(4.16) div (cid:0) Ω AD (cid:1) ∗ = 0 . That is, we claim that for any ϕ ∈ C ∞ c ( R ) we have0 = lim ℓ →∞ Z R Z R (cid:16) Ω A ℓ D,ℓ (cid:17) ∗ ϕ ( x ) − ϕ ( y ) | x − y | d x d y | x − y | = Z R Z R (cid:0) Ω AD (cid:1) ∗ ϕ ( x ) − ϕ ( y ) | x − y | d x d y | x − y | . We write Z R Z R (cid:16) Ω A ℓ D,ℓ (cid:17) ∗ ϕ ( x ) − ϕ ( y ) | x − y | d x d y | x − y | − Z R Z R (cid:0) Ω AD (cid:1) ∗ ϕ ( x ) − ϕ ( y ) | x − y | d x d y | x − y | = Z R Z R ( A ( y )Ω D ( y, x ) − A ℓ ( y )Ω D,ℓ ( y, x )) d ϕ ( x, y ) d x d y | x − y | + Z R Z R (cid:16) d A ( y, x ) − d A ℓ ( y, x ) (cid:17) d ϕ ( x, y ) d x d y | x − y | . (4.17)As for the second term of (4.17) we observe that by weak convergence of d A ℓ in L ( V od R )we have lim ℓ →∞ Z R Z R (cid:16) d A ( y, x ) − d A ℓ ( y, x ) (cid:17) d ϕ ( x, y ) d x d y | x − y | = 0 . As for the first term of (4.17) we proceed exactly as in Step 1 and obtainlim ℓ →∞ Z R Z R ( A ( y )Ω D ( y, x ) − A ℓ ( y )Ω D,ℓ ( y, x )) d ϕ ( x, y ) d x d y | x − y | = 0 . FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 21
This finishes the proof of (4.16).
Step 4.
We claim that (up to a subsequence)(4.18) lim ℓ →∞ A ℓ Ω D c ,ℓ · d u ℓ ψ = Z R A Ω D c · d uψ, where Ω D c = Ω − Ω D and Ω ∈ L ( V od R ) is the one given in the assumptions of the theorem.Indeed, since Ω D c ,ℓ ( x, y ) = 0 whenever both x, y ∈ D we have by the support of ψ , Z R A ℓ Ω D c ,ℓ · d uψ = Z R Z R ( A ℓ ( x )) ij (Ω D c ,ℓ ) jk ( x, y ) (cid:0) u kℓ ( x ) − u kℓ ( y ) (cid:1) | x − y | ψ ( x ) χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = Z R Z R (Ω D c ,ℓ ) jk ( x, y )( A ℓ ( x )) ij (cid:0) u kℓ ( x ) − u kℓ ( y ) (cid:1) | x − y | ψ ( x ) χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | . (4.19)So if we set F ℓ ( x, y ) := χ | x − y |≥ dist ( D ,∂D ) ( u ℓ ( x ) − u ℓ ( y )) | x − y | A ℓ ( x ) ψ ( x )and F ( x, y ) := χ | x − y |≥ dist ( D ,∂D ) ( u ( x ) − u ( y )) | x − y | A ( x ) ψ ( x ) . We claim that we have the strong convergence(4.20) lim ℓ →∞ k F ℓ − F k L ( V od R ) = 0 . Indeed, we have Z R Z R | F ℓ ( x, y ) − F ( x, y ) | d x d y | x − y |≤ Z R Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) A ℓ ( x ) − d u ( x, y ) A ( x ) (cid:12)(cid:12)(cid:12) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | - Z R Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | + Z R Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | . (4.21) For the first term of the right-hand side of (4.21) we take R ≫
1, such that in particularsupp ψ ⊂ D ⊂⊂ D ⊂ B ( R ) and estimate Z R Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = Z R \ B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | + Z B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | . (4.22)Now, for the second term of the right-hand side of (4.22) we have Z B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | - (cid:0) k A k L ∞ ( D ) + k A ℓ k L ∞ ( D ) (cid:1) k ψ k L ∞ Z B ( R ) \ D Z D | u ℓ ( x ) − u ( x ) | + | u ℓ ( y ) − u ( y ) | | x − y | d x d y - (cid:0) k A k L ∞ ( D ) + k A ℓ k L ∞ ( D ) (cid:1) k ψ k L ∞ dist − ( D , ∂D ) (cid:18) Z B ( R ) \ D Z D | u ℓ ( x ) − u ( x ) | d x d y + Z B ( R ) \ D Z D | u ℓ ( y ) − u ( y ) | d x d y (cid:19) ≤ C ( D , D, R ) (cid:0) k A k L ∞ ( D ) + k A ℓ k L ∞ ( D ) (cid:1) k u ℓ − u k L ( B ( R )) . Thus, by the strong convergence on compact sets of u ℓ in L we obtain(4.23)lim ℓ →∞ Z B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = 0 . Now we estimate the first term of the right-hand side of (4.22). We observe that for alllarge R , whenever x ∈ supp ψ and y B ( R ), we have | x − y | % | y | . Therefore, Z R \ B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | - (cid:0) k A k L ∞ ( D ) + k A ℓ k L ∞ ( D ) (cid:1) k ψ k L ∞ Z R \ B ( R ) Z D | u ℓ ( x ) − u ( x ) | + | u ℓ ( y ) − u ( y ) | | y | d x d y - (cid:0) k A k L ∞ ( D ) + k A ℓ k L ∞ ( D ) (cid:1) k ψ k L ∞ k u ℓ − u k L ( D ) Z R \ B ( R )
11 + | y | d y + (cid:0) k A k L ∞ ( D ) + k A ℓ k L ∞ ( D ) (cid:1) k ψ k L ∞ k u ℓ − u k L + L ∞ ( R ) max (cid:26)Z R \ B ( R )
11 + | y | d y,
11 + R (cid:27) - R − (cid:0) k A k L ∞ ( D ) + k A ℓ k L ∞ ( D ) (cid:1) k ψ k L ∞ k u ℓ − u k L + L ∞ ( R ) . FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 23
Thus,(4.24)lim R →∞ sup ℓ Z R \ B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) (cid:0) | A ( x ) | + | A ℓ ( x ) | (cid:1) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = 0 . Combining (4.22) with (4.23) and (4.24) we obtain the convergence of the first term of theright-hand side of (4.21)(4.25) lim ℓ →∞ Z R Z D (cid:12)(cid:12)(cid:12) d u ℓ ( x, y ) − d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = 0 . As for the second term of the right-hand side of (4.21) we estimate for an R ≫ Z R Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = Z B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | + Z R \ B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | . (4.26)For the first term of (4.26) we observe that since A ℓ → A pointwise almost everywhere,then we havelim ℓ →∞ (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) | x − y | = 0 pointwise a.e. in D × B ( R ) . Moreover, we have (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) | x − y | - (cid:18) sup ℓ k A ℓ k L ∞ + k A k L ∞ (cid:19) (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) | x − y | and the right-hand side is independent of ℓ and integrable. Thus, by dominated convergencetheorem we have(4.27) lim ℓ →∞ Z B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = 0 . For the second term of (4.26) we observe that for large R whenever x ∈ supp ψ and y B ( R ), we have | x − y | % | y | . Z R \ B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | - R − k ψ k L ∞ (cid:0) k A ℓ k L ∞ ( D ) + k A k L ∞ ( D ) (cid:1) k u k L + L ∞ ( R ) Thus,lim R →∞ sup ℓ Z R \ B ( R ) Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = 0 , which combined with (4.26) and (4.27) gives(4.28) lim ℓ →∞ Z R Z D (cid:12)(cid:12)(cid:12) d u ( x, y ) (cid:12)(cid:12)(cid:12) | A ℓ ( x ) − A ( x ) | | ψ ( x ) | χ | x − y |≥ dist ( D ,∂D ) d x d y | x − y | = 0 . Now, plugging (4.28) and (4.25) into (4.21) we establish (4.20).Thus, (4.20) and a combination of the weak convergence of Ω ℓ,D c and the strong convergenceof F ℓ implies lim ℓ →∞ Z R Ω ℓ,D c ( x, y ) F ℓ ( x, y ) d x d y | x − y | = Z R Ω D c ( x, y ) F ( x, y ) d x d y | x − y | . This establishes (4.18).
Step 5.
We claim that(4.29) lim ℓ →∞ f ℓ [ A ℓ ψ ] = f [ Aψ ] . Indeed, this holds because A ℓ ψ is uniformly bounded in W , and by assumption f ℓ → f in ( W , ) ∗ . Step 6.
Passing to the limit.Passing with ℓ → ∞ in (4.5), using (4.7), (4.15), (4.18), and (4.29), we obtain(4.30) Z R Ad u · d ψ = Z R (cid:0) Ω AD (cid:1) ∗ · ud ψ + Z R A Ω D c · d u ψ + f [ Aψ ] . By (4.15) we know that (cid:0) Ω AD (cid:1) ∗ is -divergence free and thus by Lemma 2.1 we have Z R (cid:0) Ω AD (cid:1) ∗ · ud ψ = Z R Ω AD · d uψ, which combined with (4.30) and formulas Ω AD = A Ω D − d A and Ω D c = Ω − Ω D gives(4.31) Z R Ad u · d ψ = Z R Ω A · d uψ + f [ Aψ ] . This holds for any ψ ∈ C ∞ c ( D ). By density we can plug in ψ := A − ϕ for any ϕ ∈ C ∞ c ( D ),which, by Lemma 4.1, leads to the claim. (cid:3) Corollary 4.3.
Let u ℓ , Ω ℓ , and f ℓ be as in Theorem 1.3. Let D ⊂ R . Then there exits alocally finite Σ ⊂ D such that ( − ∆) u = Ω · d u + f in D \ Σ . FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 25
Proof.
We follow in spirit the covering argument of Sacks–Uhlenbeck [29, Proposition 4.3& Theorem 4.4].By assumptions there is a number Λ > ℓ ∈ N k Ω ℓ k L ( V od R ) < Λ.Let α ∈ N and define B α := { B ( x i,α , − α ) : x i,α ∈ D } be a family of balls such that D ⊂ S B α and each point x ∈ D is covered at most λ times, and such that for a smallerradius we still have D ⊂ S i B ( x i,α , − α − ). Then X i Z B ( x i,α , − α ) Z R | Ω ℓ ( x, y ) | d x d y | x − y | < Λ λ. Now, let σ > Λ λσ balls in B α on which Z B ( x i,α , − α ) Z R | Ω ℓ ( x, y ) | d x d y | x − y | > σ. Thus, by Proposition 4.2, we obtain that except for
K < Λ λσ + 1 balls from B α we have(4.32) Z R d u · d ϕ i = Z R Ω · d uϕ i + f [ ϕ i ] for all ϕ i ∈ C ∞ c ( B ( x i,α , − α − )) . Let us denote those balls by B ( y i,α , − α ) for i = 1 , . . . , K . Then by (4.32) we get(4.33) Z R d u · d ψ = Z R Ω · d uψ + f [ ψ ] , for all ψ ∈ C ∞ c ( D \ [ i ≤ K B ( y i,α , − α − )) . Since S α ∈ N (cid:16) D \ S Ki =1 B ( y i,α , − α − ) (cid:17) = D \ { x , . . . , x K } gives (4.33) for any ψ ∈ C ∞ c ( D \ Σ), where Σ := { x , . . . , x K } . (cid:3) In order to conclude we will need a removability of singularities lemma, compare with [18,Proposition 4.7].
Lemma 4.4.
Let u ∈ ˙ W , ( R , R N ) , f ∈ L ( R , R N ) , and g ∈ W − , ( R ) . Assume that forsome locally finite set Σ ⊂ D we have ( − ∆) u = f + g in D \ Σ . Then ( − ∆) u = f + g in D. Proof.
For simplicity of presentation let us assume that Σ = { x } . By definition we havefor any ϕ ∈ C ∞ c ( D \ { x } ) Z D Z D ( u ( x ) − u ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d x d y = Z D f ( x ) ϕ ( x ) d x + g [ ϕ ] . Let { ζ ℓ } ℓ ∈ N ⊂ C ∞ c ( D, [0 , ℓ ∈ N we have(4.34) ζ ℓ ≡ B ρ ℓ ( x ) , ζ ℓ ≡ B R ℓ ( x ) , and lim ℓ →∞ [ ζ ℓ ] W , ( D ) = 0for a 0 < ρ ℓ < R ℓ → ℓ → ∞ .Now let ψ ∈ C ∞ c ( D ) and then ψ ℓ := ψ (1 − ζ ℓ ) ∈ C ∞ c (Σ \ { x } ) is an admissible test functionand we have(4.35) Z D Z D ( u ( x ) − u ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y −I ℓ = Z D f ( x ) ψ ( x ) d x + g [ ψ ] −II ℓ −III ℓ . We have I ℓ := Z D Z D ( u ( x ) − u ( y ))( ψ ( x ) ζ ℓ ( x ) − ψ ( y ) ζ ℓ ( y )) | x − y | d x d y = Z D Z D ( u ( x ) − u ( y )) ψ ( x )( ζ ℓ ( x ) − ζ ℓ ( y )) | x − y | d x d y + Z D Z D ( u ( x ) − u ( y ))( ψ ( x ) − ψ ( y )) ζ ℓ ( y ) | x − y | d x d y ≤ k ψ k L ∞ ( D ) [ u ] W , ( D ) [ ζ ℓ ] W , ( D ) + Z B Rℓ Z D ( u ( x ) − u ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y, (4.36)Thus, by (4.34) and by the absolute continuity of the integral we have lim ℓ →∞ I ℓ = 0.Secondly,(4.37) II ℓ := Z D f ( x ) ψ ( x ) ζ ℓ ( x ) d x ≤ k ψ k L ∞ Z B Rℓ | f ( x ) | d x ℓ →∞ −−−→ , by the absolute continuity of the integral.Thus, passing with ℓ → ∞ in (4.35) we get for any ψ ∈ C ∞ c ( D ) Z D Z D ( u ( x ) − u ( y ))( ψ ( x ) − ψ ( y )) | x − y | d x d y = Z D f ( x ) ψ ( x ) d x. Lastly,
III ℓ := g [ ψ ζ ℓ ] ℓ →∞ −−−→ , because, by (4.34), we have [ ψ ζ ℓ ] W , ℓ →∞ −−−→ (cid:3) Proof of Theorem 1.3.
Combining Corollary 4.3 and Lemma 4.4 we obtain the claim. (cid:3)
FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 27
Appendix A. Nonlocal Hodge decomposition
Lemma A.1.
Let p > , s ∈ (0 , , G ∈ L p ( V od R n ) then there exists a unique decompo-sition G = d s a + B, where a ∈ ˙ W s,p ( R n ) with lim | x |→∞ a ( x ) = 0 and B ∈ L p ( V od R n ) with div s B = 0 . More-over, (A.1) k B k L p ( V od R n ) + [ a ] W s,p ( R n ) - k G k L p ( V od R n ) . Proof.
Since G ∈ L p ( V od R n ) we have div s G ∈ (cid:0) W s,p ′ ( R n ) (cid:1) ∗ , namelydiv s G [ ϕ ] - k G k L p ( V od R n ) [ ϕ ] W s,p ′ ( R n ) . We have div s G ∈ ˙ F − sp,p , since ( − ∆) − s : ˙ F − sp,p ( R n ) → ˙ F sp,p ( R n ) is an isomorphism [28, § a ∈ ˙ F sp,p ( R n ) to thedistributional equation ( − ∆) s a = div s G. with [ a ] ˙ F sp,p ( R n ) - [div s G ] F − sp ′ ,p ′ ( R n ) - k G k L p ( V od R n ) . Since s ∈ (0 , F sp,p ( R n ) is a function space modulo constants, and we can fix the constantby assuming with lim | x |→∞ a ( x ) = 0. Then we have found a ∈ ˙ F sp,p ( R n ) = ˙ W s,p ( R n ), andwe have Z R n d s a · d s ϕ = Z R n F ϕ ∀ ϕ ∈ C ∞ c ( R n ) . The uniqueness of a would also follow by considering a difference of two solutions and anapplication of nonlocal Liouville theorem [11, Theorem 1.1].Now define B := G − d s a . We havediv s B = div s G − div s ( d s a ) = div s G − ( − ∆) s a = 0 , which finishes the proof. (cid:3) Appendix B. Localization
The following follows from a relatively straight-forward localization results, see e.g. [19].
Proposition B.1.
Assume D ⊂⊂ D ⊂⊂ D ′ ⊆ D ⊆ R open intervals and let u ∈ L ( R , R N ) + L ∞ ( R , R N ) ∩ ˙ W s, ( D, R N ) be a solution to ( − ∆) D u = Ω · D d u + f in D ′ . That is, assume Z D Z D ( u ( x ) − u ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d x d y = Z D Z D Ω( x, y ) d u ( x, y ) ϕ ( x ) d x d y | x − y | + Z D f ϕ, ∀ ϕ ∈ C ∞ c ( D ′ ) . (B.1) Let η ∈ C ∞ c ( D ) and set v := ηu and ˜Ω ij ( x, y ) = χ D ( x ) χ D ( y )Ω ij ( x, y ) . Then ( − ∆) v = ˜Ω · d v + ηf + G ( u, · ) in R , where G is a bilinear form with the following estimates for any s ∈ (0 , ) and ε > |G ( u, · ) | ≤ C ( η, s, ε, D , D ) (cid:16) k Ω k L ( V od D ) (cid:17) · (cid:0) k u k L ( D )+ L ∞ ( D ) + [ u ] W s, ( D ) (cid:1) · (cid:16) k ϕ k L ( D )+ L ∞ ( D ) + k ϕ k L s ( D ) + k ϕ k L + L ∞ ( R ) + [ ϕ ] W ε, s +1 ( D ) (cid:17) . In particular we have k ˜Ω k L ( V od R ) ≤ C k Ω k L ( V od D ) . Proof.
Let ϕ ∈ C ∞ c ( R ). We have( η ( x ) u ( x ) − η ( y ) u ( y )) ( ϕ ( x ) − ϕ ( y ))= ( u ( x ) − u ( y ))( η ( x ) ϕ ( x ) − η ( y ) ϕ ( y )) + ( η ( x ) − η ( y )) ( u ( y ) ϕ ( x ) − u ( x ) ϕ ( y )) . Since ηϕ ∈ C ∞ c ( D ′ ) it is an admissible test function and we have from the equation (B.1) Z D Z D ( v ( x ) − v ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d x d y = Z D Z D Ω( x, y ) d u ( x, y ) η ( x ) ϕ ( x ) d x d y | x − y | + Z R f ηϕ + G ( u, ϕ ) . Here, G ( u, ϕ ) = Z D Z D ( η ( x ) − η ( y )) ( u ( y ) ϕ ( x ) − u ( x ) ϕ ( y )) | x − y | d x d y. Moreover, we have Z R Z R ( v ( x ) − v ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d x d y = Z D Z D ( v ( x ) − v ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d x d y + G ( u, ϕ ) , where, because supp v ⊂ D , G ( u, ϕ ) = 2 Z D v ( x ) Z R \ D ( ϕ ( x ) − ϕ ( y )) | x − y | d x d y. FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 29
That is we have Z R Z R ( v ( x ) − v ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d x d y = Z D Z D Ω( x, y ) d u ( x, y ) η ( x ) ϕ ( x ) d x d y | x − y | + Z R f ηϕ + G ( u, ϕ ) + G ( u, ϕ ) . Furthermore, since d u ( x, y ) η ( x ) = d ( ηu )( x, y ) − u ( y ) d η ( x, y )and supp v ⊂ D , we have Z D Z D Ω( x, y ) d u ( x, y ) η ( x ) ϕ ( x ) d x d y | x − y | = Z D Z D Ω( x, y ) d v ( x, y ) ϕ ( x ) d x d y | x − y | − Z D Z D Ω( x, y ) u ( y ) d η ( x, y ) ϕ ( x ) d x d y | x − y | = Z R Z R χ D ( x ) χ D ( y )Ω( x, y ) d v ( x, y ) ϕ ( x ) d x d y | x − y | + Z D \ D Z D Ω( x, y ) d v ( x, y ) ϕ ( x ) d x d y | x − y | + Z D Z D \ D Ω( x, y ) d v ( x, y ) ϕ ( x ) d x d y | x − y |− Z D Z D Ω( x, y ) u ( y ) d η ( x, y ) ϕ ( x ) d x d y | x − y | . So if we set G ( u, ϕ ) := Z D \ D Z D Ω( x, y ) d v ( x, y ) ϕ ( x ) d x d y | x − y | + Z D Z D \ D Ω( x, y ) d v ( x, y ) ϕ ( x ) d x d y | x − y | and G ( u, ϕ ) := − Z D Z D Ω( x, y ) u ( y ) d η ( x, y ) ϕ ( x ) d x d y | x − y | , then we have shown for any ϕ ∈ C ∞ c ( R ), Z R Z R ( v ( x ) − v ( y ))( ϕ ( x ) − ϕ ( y )) | x − y | d x d y = Z R ˜Ω · d v ϕ + Z R f ηϕ + X i =1 G i ( u, ϕ ) . It remains to estimate each G i ( u, ϕ ).Estimate of G : By the support of η we have G ( u, ϕ ) = Z D Z D ( η ( x ) − η ( y )) ( u ( y ) ϕ ( x ) − u ( x ) ϕ ( y )) | x − y | d x d y + 2 Z D Z D \ D ( η ( x ) − η ( y )) ( u ( y ) ϕ ( x ) − u ( x ) ϕ ( y )) | x − y | d x d y. (B.2) As for the first term we have Z D Z D ( η ( x ) − η ( y )) ( u ( y ) ϕ ( x ) − u ( x ) ϕ ( y )) | x − y | d x d y ≤ k η k Lip Z D Z D | u ( y ) ϕ ( x ) − u ( x ) ϕ ( y ) || x − y | d x d y - k η k Lip (cid:18)Z D | u ( y ) | Z D | ϕ ( x ) − ϕ ( y ) || x − y | d x d y + Z D | ϕ ( y ) | Z D | u ( x ) − u ( y ) || x − y | d x d y (cid:19) - k η k Lip Z D | u ( y ) − ( u ) D | Z D | ϕ ( x ) − ϕ ( y ) || x − y | d x d y + k η k Lip k u k L ( D ) Z D Z D | ϕ ( x ) − ϕ ( y ) || x − y | d x d y + k η k Lip Z D | ϕ ( y ) | Z D | u ( x ) − u ( y ) || x − y | d x d y. (B.3)We observe that for any p ∈ (1 , ∞ ) and any ε > Z D (cid:18)Z D | ϕ ( x ) − ϕ ( y ) || x − y | d x (cid:19) p d y = Z D (cid:18)Z D | ϕ ( x ) − ϕ ( y ) || x − y | ε | x − y | ε d x | x − y | (cid:19) p d y - [ ϕ ] W ε,p ( D ) Z D (cid:18)Z D | x − y | εp ′ d x | x − y | (cid:19) pp ′ d y - C ( D )[ ϕ ] W ε,p ( D ) . Thus, for any ε > s ∈ (0 , ) we have Z D | u ( y ) − ( u ) D | Z D | ϕ ( x ) − ϕ ( y ) || x − y | d x d y + k u k L ( D ) Z D Z D | ϕ ( x ) − ϕ ( y ) || x − y | d x d y - C ( D ) (cid:16) k u − ( u ) D k L − s ( D ) [ ϕ ] W ε, s +1 ( D ) + k u k L ( D ) [ ϕ ] W ε, s +1 ( D ) (cid:17) . (B.4)We also have(B.5) Z D | ϕ ( y ) | Z D | u ( x ) − u ( y ) || x − y | d x d y - k ϕ k L ( D ) [ u ] W s, ( D ) . Combining (B.3) with (B.4) and (B.5) we obtain Z D Z D ( η ( x ) − η ( y )) ( u ( y ) ϕ ( x ) − u ( x ) ϕ ( y )) | x − y | d x d y - k η k Lip (cid:0) k u k L ( D ) + [ u ] W s, ( D ) (cid:1) (cid:16) k ϕ k L ( D ) + [ ϕ ] W ε, s +1 ( D ) (cid:17) . (B.6) FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 31
For the second term of (B.2) observe that for x ∈ D and y ∈ D \ D we have | x − y | ≈ | y | ,so we have 2 (cid:12)(cid:12)(cid:12)(cid:12)Z D Z D \ D ( η ( x ) − η ( y )) ( u ( y ) ϕ ( x ) − u ( x ) ϕ ( y )) | x − y | d x d y (cid:12)(cid:12)(cid:12)(cid:12) - k η k L ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z D Z D \ D | u ( y ) | | ϕ ( x ) | + | u ( x ) | | ϕ ( y ) | | y | d x d y (cid:12)(cid:12)(cid:12)(cid:12) - k η k L ∞ k u k L + L ∞ ( D ) k ϕ k L + L ∞ ( D ) . (B.7)Thus, by (B.2), (B.6), and (B.7) we get(B.8) |G ( u, ϕ ) | - (cid:0) k u k L + L ∞ ( D ) + [ u ] W s, ( D ) (cid:1) (cid:16) k ϕ k L + L ∞ ( D ) + [ ϕ ] W ε, s +1 ( D ) (cid:17) . Estimate of G : Similarly as in (B.7), if x ∈ D and y ∈ R \ D we have | x − y | ≈ | y | ,and thus |G ( u, ϕ ) | - k ηu k L ( D ) (cid:0) k ϕ k L ( D ) + k ϕ k L ( R ) (cid:1) - k u k L ( D ) (cid:0) k ϕ k L ( D ) + k ϕ k L + L ∞ ( R ) (cid:1) . Estimate of G : Using the support of v , observing again that | x − y | % | y | if y ∈ R \ D and x ∈ D . |G ( u, ϕ ) | - k Ω k L ( V od D ) (cid:18)Z D \ D Z D | u ( x ) | | ϕ ( x ) | d x d y | y | + Z D Z D \ D | u ( y ) | | ϕ ( x ) | d x d y | x | (cid:19) - k Ω k L ( V od D ) (cid:0) k uϕ k L ( D ) + k ϕ k L ( D ) k u k L ( D ) (cid:1) - k Ω k L ( V od D ) (cid:16) k u k L ( D ) k ϕ k L ( D ) + k u − ( u ) D k L − s ( D ) k ϕ k L s ( D ) + k ϕ k L ( D ) k u k L ( D ) (cid:17) - k Ω k L ( V od D ) (cid:16) k u k L ( D ) k ϕ k L ( D ) + [ u ] W s, ( D ) k ϕ k L s ( D ) + k ϕ k L ( D ) k u k L ( D ) (cid:17) - k Ω k L ( V od D ) (cid:0) k u k L ( D ) + [ u ] W s, ( D ) (cid:1) (cid:16) k ϕ k L s ( D ) + k ϕ k L ( D ) (cid:17) . This argument works for any s ∈ (0 , ).Estimate of G : We have |G ( u, ϕ ) | - k Ω k L ( V od D ) (cid:18)Z D Z D | u ( y ) d η ( x, y ) ϕ ( x ) | d x d y | x − y | (cid:19) . Now observe that | d η ( x, y ) | ≤ k η k | x − y | , thus (cid:18)Z D Z D | u ( y ) d η ( x, y ) ϕ ( x ) | d x d y | x − y | (cid:19) - k u k L ( D ) k ϕ k L ( D ) . On the other hand (cid:18)Z D Z D | u ( y ) d η ( x, y ) ϕ ( x ) | d x d y | x − y | (cid:19) - [ η ] W , k u k L ∞ ( D ) k ϕ k L ∞ ( D ) . We also have (cid:18)Z D Z D | u ( y ) d η ( x, y ) ϕ ( x ) | d x d y | x − y | (cid:19) - k u k L ∞ ( D ) k ϕ k L ( D ) sup x ∈ D (cid:18)Z D | η ( x ) − η ( y ) | | x − y | d y (cid:19) and sup x ∈ D (cid:18)Z D | η ( x ) − η ( y ) | | x − y | d y (cid:19) - k η k Lip . Thus combining the estimates on G we obtain |G ( u, ϕ ) | - k Ω k L ( V od D ) k u k L + L ∞ ( D ) k ϕ k L + L ∞ ( D ) . (cid:3) Appendix C. A Sobolev inequality
Theorem C.1.
Let s ∈ (0 , , p, q ∈ (1 , ∞ ) and f ∈ L p ( R n ) then (1) [ f ] ˙ F sp,q ( R n ) - [ f ] W sp,q ( R n ) ;(2) if p > nqn + sq then [ f ] W sp,q ( R n ) - [ f ] ˙ F sp,q ( R n ) . The constants depend on s, p, q, n and are otherwise uniform.
While characterizations such as Theorem C.1 are well-known for Besov spaces, for Triebelspaces this seems to have been known only for q = p (where it follows from the Besov-spacecharacterization), q = 2 where it is a result due to Stein and Fefferman, [31, 12]. It wasalso known “for large s” [32, Section 2.5.10]. Also a conjecture Theorem C.1 holds is verynatural, quite surprisingly, to the best of our knowledge the first time Theorem C.1 hasbeen proven is very recently by Prats [25]. Corollary C.2.
Let s ∈ (0 , , t ∈ ( s, and p, p ∗ ∈ (1 , ∞ ) where s − np ∗ = t − np . If q ∈ (1 , ∞ ) such that p ∗ > nqn + sq we have k|D s,q f |k L p ∗ ( R n ) - k ( − ∆) t f k L p ( R n ) . More precisely, in terms of Lorentz spaces we have for any r ∈ [1 , ∞ ] , k|D s,q f |k L ( p ∗ ,r ) ( R n ) - k ( − ∆) t f k L ( p,r ) ( R n ) . FRACTIONAL VERSION OF RIVI`ERE’S GL(N)-GAUGE 33
Proof.
From Theorem C.1 we have k|D s,q f |k L p ∗ ( R n ) ≈ [ f ] F sp ∗ ,q ( R n ) . We recall the Sobolev-embedding theorem for Triebel-Lizorkin spaces ˙ F tp, ˜ q ֒ → ˙ F tp,q for any q, ˜ q ∈ (1 , ∞ ). Thus, k|D s,q f |k L p ∗ ( R n ) - [ f ] F tp, ( R n ) - k ( − ∆) t f k L p ( R n ) . As for the Lorentz-space estimate we can argue by real interpolation. Indeed, fix s, q, p, p ∗ .Observe that f
7→ |D s,q f | is a sublinear operator.We can find p < p < p such that p and p are still admissible, and thus we have k|D s,q f |k L p ∗ i ( R n ) - [ f ] F tp, ( R n ) - k ( − ∆) t f k L pi ( R n ) i = 1 , . From real interpolation we now obtain the Lorentz space claim. (cid:3)
Appendix D. A sequence of cut-off functions in the critical Sobolevspace
For readers convenience we present here a proof of a well known result, which essentiallysays that in the critical Sobolev space a point has zero capacity. See for example [1,Theorem 5.1.9], compare also with a similar construction [23, Lemma 3.2].
Lemma D.1.
There exists a sequence of functions with the following properties: { ζ ℓ } ℓ ∈ N ⊂ C ∞ c ( R , [0 , and for all ℓ ∈ N we have (D.1) ζ ℓ ≡ on B ρ ℓ ( x ) , ζ ℓ ≡ outside B R ℓ ( x ) , and lim ℓ →∞ [ ζ ℓ ] W , ( R ) = 0 for a sequence of radii < ρ ℓ < R ℓ → as ℓ → ∞ .Proof. Let f ( x ) = log log (cid:16) | x | (cid:17) ∈ W , ( B , R ) be an unbounded function. We define˜ Z k ( x ) := f ( x ) ≥ k + 1 ,f ( x ) − k if k ≤ f ( x ) ≤ k + 1 , f ( x ) < k. Then, ∇ ˜ Z k ( x ) := f ( x ) ≥ k + 1 , ∇ f ( x ) if k ≤ f ( x ) ≤ k + 1 , f ( x ) < k. The support of ∇ ˜ Z k is the set B k := (cid:8) x ∈ B : A k +1 ≤ | x | ≤ A k (cid:9) , where A k = r e e k − , A k +1 ≤ A k , and lim k →∞ A k = 0 . Now, Z B |∇ ˜ Z k | d x = Z A k +1 ≤| x |≤ A k |∇ ˜ Z k | d x k →∞ −−−→ , which follows from the fact that ∇ ˜ Z k ∈ L ( B ) and that |{ x ∈ B : A k +1 ≤ | x | ≤ A k }| shrinks to zero.Thus, we obtained a sequence of functions for which˜ Z k ≡ B A k +1 , ˜ Z k ≡ B A k , and lim k →∞ k∇ ˜ Z k k L ( B ) = 0 . By extending by zero we obtain a sequence Z k ∈ W , ( R ) with the properties(D.2) Z k ≡ B A k +1 , Z k ≡ B A k , and lim k →∞ k∇ Z k k L ( R ) = 0 . Defining now ζ k := Z k (cid:12)(cid:12) R in the trace sense we obtain by the trace inequality, [13][ ζ k ] W , ( R ) - k∇ Z k k L ( R ) k →∞ −−−→ . Approximating { ζ k } k ∈ N by smooth functions we obtain the desired sequence. (cid:3) References [1] D. R. Adams and L. I. Hedberg.
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Department of Mathematics, ETH Z¨urich, R¨amistrasse 101, 8092 Z¨urich,Switzerland
Email address : [email protected] (Katarzyna Mazowiecka) Universit´e catholique de Louvain, Institut de Recherche enMath´ematique et Physique, Chemin du Cyclotron 2 bte L7.01.02, 1348 Louvain-la-Neuve,Belgium
Email address : [email protected] (Armin Schikorra) Department of Mathematics, University of Pittsburgh, 301 ThackerayHall, Pittsburgh, PA 15260, USA
Email address : [email protected] (Lifeng Wang) Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall,Pittsburgh, PA 15260, USA
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