UNIQUENESS

. Extending an argument by Shatah and Struwe we obtain uniqueness for solutions of the half-wave map equation in dimension d ≥ 3 in the natural energy class.


Introduction and main result
Half-wave maps appear in the physics literature as the continuum limit of Calagero-Moser spin systems, see [13] and references within.They are solutions u : R d × [0, T ] → S 2 ⊂ R 3 to the half-wave maps equation which is given by (1.1) Here and henceforth ∧ denotes the cross product in R 3 and (−∆) 1 2 = |∇| is the half-Laplacian.Recently, several authors, e.g.[10,11,3,1,9,20,14], began to study mathematical properties of (1.1).In [10] the structural relation of the half-wave map equation to the wave map equation ∂ tt u − ∆u = (−∂ t u • ∂ t u + ∇u • ∇u) u was discovered and exploited.Namely, by a direct computation, see [10, p.663], a solution of (1.1) solves The authors of [10] then raised the question if one can use this route to extend methods developed for wave maps, e.g.those in the celebrated articles [23,21,22], to half-wave maps.Following this principle, in [10,9,14] different well-posedness results for large dimensions were discovered.Observe that the energy-critical dimension for the halfwave map equation is d = 1, as opposed to the energy-critical dimension of the wave map equation, which is d = 2.In this work we also follow this spirit of treating solutions to the halfwave map equation as solutions to a wave-map-type equation, but we focus on techniques developed for wave maps by Shatah and Struwe [19].Our main result is the following uniqueness property of half-wave maps.
) for some Q ∈ S 2 , and if Here L (p,q) denotes the Lorentz space.The a priori assumptions (1.3) are the natural energy assumptions for initial data u 0 , v 0 ∈ Ḣ d 2 (R d ), which was one of the crucial observations in [19] where Shatah and Struwe observed this for α = 1.A careful inspection of their argument actually gives the assumption (1.3) for small α > 1, see Section 5.As in the case of Shatah-Struwe, our arguments rely mostly on geometric properties combined with fractional Leibniz rules and related commutator estimates.However, while for the wave map equation the proof of uniqueness fits on one page, our argument does not -since it relies on several further structural observations of the "tangential part" of the right-hand side of (1.2), which we hope are of independent interest.

Outline.
In Section 2 we introduce operators and estimates needed in the proof of Theorem 1.1.We believe that most, if not all, of these estimates are known at least to some experts -and they can be proven by standard techniques.In Section 3 we discuss the main part of the proof, the decay estimates in time, Theorem 3.1.While we are substantially inspired by the argument by Shatah-Struwe, our estimates are more elaborate, even though they mostly rely on the fractional Leibniz rule.The decay estimates of Theorem 3.1 combined the standard Grönwall type inequality imply Theorem 1.1, see Section 4. In Section 5 we discuss the suitability of the assumptions (1.3) for α > 1, α ≈ 1.We believe that our arguments can also be used to discuss existence for small data in the above energy class as in Shatah-Struwe, which will be the subject of a future investigations.
The fractional Laplacian is as a multiplier operator via the Fourier transform F for a constant c > 0, We also remark the useful potential representation for some (different) constant c ∈ R and s ∈ (0, 1) |x − y| n+s dy, and, for s ∈ (0, 2), |h| d+s dh.
As for negative powers, I s ≡ (−∆) − s 2 denotes the Riesz potential, ).It has the potential representation for s ∈ (0, d), Some of our arguments will depend on Lorentz space estimate, L p,q (R d ).We only recall the main properties and refer the reader to [4,Section 1.4]: ) is often referred to as the weak L p -space.

Embedding theorems.
A casual observation we will use throughout this paper is the following comparability Lemma 2.1.For any p ∈ (1, ∞),

More generally in the realm of Lorentz spaces, for any
Proof.This follows since the Riesz transforms combined with the following facts that can easily be checked using the Fourier transform, Equivalently, in terms of the Riesz potential I α ≡ |∇| −α we have In terms of Lorentz spaces we have for any q ∈ [1, ∞], and An important limit version is .
Assume now β ∈ (0, 1  2 ] and p ∈ 2.2.Leibniz rule commutators.In the following we discuss mostly Leibniz rule type estimates.The Leibniz rule operator for |∇| s will be denoted by As a standing assumption, we are going to assume that all functions belong to . By density arguments we can apply these inequalities to the situation in the next section.Let us stress that we make no effort to obtain the sharpest possible result with respect to L p -spaces (in particular we generally rule out p = 1 and p = ∞) but instead focus on the applicability for our purposes.By a direct computation we have the following useful formula, which has been observed by many authors.Lemma 2.5.Let s ∈ (0, 2) then for some c = c(s, n), We now begin by stating several useful estimates for the Leibniz rule operator, most of them are probably known to some experts -and all of them can be proven via standard methods.
We can also estimate a differentiated version of the Leibniz rule operator.
. We can conclude.
We will also need an estimate for a double commutator.We are not aware of this estimate in the literature, but it can be obtained with the usual paraproduct approach.
With the usual paraproduct argument, denoting by ∆ ℓ := l≤ℓ ∆ l, (2.4) Here, by a slight abuse of notation we say for indices ℓ, k that ℓ ≈ k if k−c ≤ ℓ ≤ k+c for some constant c > 0.
The last term in (2.4) is the simplest to estimate, since for any The other terms are very similar to each other.We only discuss the first term.By Plancherel theorem we have ˆRd Here k is the symbol of the operator H|∇| β ,|∇| α given by We observe that by the support of the Littlewood-Paley projection operators ∆ j and ∆ j−4 , in the integral above we have |ξ − η| ≤ 1 2 |η|.By a Taylor expansion, where m ℓ (η) and n ℓ (η) are zero homogeneous functions.Now we observe that for any θ ≥ 0 In the last inequality we used the Littlewood-Paley theorem, [5, Theorem 1.
and thus we actually have With this method we can estimate each of the terms in (2.4) and obtain the claim.
The next result is very similar to the estimate of Lemma 2.7 (which indeed can be proven with the techniques of the following lemma).
We will need Leibniz-rule estimates involving three terms, the basis of which is the following Lemma.

and
(2.6) (Observe the previous assumptions are trivially satisfied if p i ≥ 2).
Then we have Proof.Since From another application of Hölder's inequality, Here the W α,p q -seminorm for α ∈ (0, 1) and p, q ∈ (1, ∞) is defined as Our choice for q i ensures p i > dqi d+αiqi , so we have by the results in [15], where Ḟ denotes the homogeneous Triebel-Lizorkin space.That is, we have Thus, we have established the claim and can conclude.
We now state a version similar to Lemma 2.10 but for α 3 < 0.
Proof.As in Lemma 2.10, since p i > 2 for i = 1, 2 we can choose q 1 = p 1 and q 2 = p 2 and set . By the same argument as in the proof of Lemma 2.10, So, with the same Hölder inequality and embedding theorems as in Lemma 2.10, . We now observe that by Sobolev inequality, Lemma 2.2, .
The above is correct as long as this is satisfied and can conclude.

Specific estimates.
In this section we record estimates for specific L p -spaces of interest.These are mostly consequences from the estimates above, and will be useful throughout the next section.Lemma 2.12.Let d ≥ 2 then (2.8) Proof of (2.7).From (2.2) we have Now the claim follows by Sobolev embedding, Lemma 2.2.
For later use we also record the following easy consequence of the Leibniz rule estimate Lemma 2.13. .
Proof.With the help of Leibniz rules and Sobolev embedding, Lemma 2.2, .
Next we record an estimate for another version of a sort of double Leibniz rule.

Decay estimate in time
In this section we prove the main estimate for Theorem 1.1 which is where for any α > 1 we can estimate It remains to prove Theorem 3.1.
3.1.Proof of Theorem 3.1.We observe So what we need to do is multiply the equation for ∂ tt w − ∆w with ∂ t w.From the equation (1.1) for u and v, respectively, we find that Here we recall the commutator notation We will prove the estimate of Theorem 3.1 by estimating each line in (3.2), which will become increasingly more challenging, the last line being the most involved estimate.Having said that, the difficulties are mostly of algebraic nature, and the actual estimates rely on the fractional Leibniz rule discussed in Section 2.

Repeating estimates
Throughout the remainder of the section we will use Lemma 2.1 implicitly -without further mentioning.Moreover, observe that for 1 2 < α 1 < α 2 < d + 1 2 we have from Sobolev embedding, Lemma 2.2, In particular for any α ∈ This will be also used frequently and implicitly -in particular to obtain the estimate in Theorem 3.1 from the lemmata below.
Estimating the first line of (3.2) We begin with the following estimate which is proven in Shatah-Struwe [19].
Estimating the Second line of (3.2) In a similar spirit to Lemma 3.2 we can also obtain Lemma 3.3.For d ≥ 3 we have Proof.We split From Hölder's inequality and Sobolev inequality, Lemma 2.2, This provides the desired estimate for the first term in (3.3).
The second and third term in (3.3) are very similar, we only estimate the second one.Here we use the trick from [19] that they used to obtain Lemma 3.2: Since u Using that u solves the half-wave map equation (1.1) and |u| ≡ 1 we conclude Thus using Hölder inequality and Sobolev inequality, Lemma 2.2, as before, Estimating the third line of (3.2) We recall our notation for the Leibniz rule operator Observe that since |u| 2 ≡ 1 we have Regarding the first term in (3.6) we observe that this is a more complicated structure to estimate, since two terms including u − v appear to the full differential order, and we are not aware of a trick in the spirit of [19] that would change that.Instead we use the commutator structure of By (2.9) we have Also the second and third term of (3.6) are relatively straight-forward to estimate using the commutator structure of H |∇| .
Lemma 3.5.For d ≥ 2, we have Proof.We only consider the first term, the second follows from the same argument.By Hölder's inequality, and (2.7), We can conclude since Estimating the last line of (3.2) We still need to understand the estimates for We observe that we can estimate the first term of (3.7) assuming a bound on |∇| α u and |∇| α v for an arbitrarily small α > 1.

Next we consider the term
We first establish the following estimate which estimates the first term on the righthand side in (3.8) Lemma 3.7.For any α ∈ (1, d + 1 2 ), and any Proof.We recall the formula (3.9) Take any α ∈ (1, d + 1 2 ) and set σ := 2 − α.Then , and by (2.2) we can estimate the first two terms of (3.9), For the third term in (3.9) observe that for fixed i, using Lemma 2.5, This implies that For the first term, just as above for (3.10), For the second term we use Lemma 2.10.Take σ, θ > 0 such that σ + θ ∈ (0, 1).Then, (if d ≥ 2 we can take σ, θ small enough to make any of the norms below finite), This establishes the right estimate for the second term in (3.9).For the last term in (3.9) it remains we consider, again using Lemma 2.5, Thus we have for i = 1, 2, 3, We estimate the first term in (3.11).Applying first (2.2), for any small σ > 0, Sobolev embedding, Lemma 2.2, and then (2.8), For the second term in (3.11), by (2.2), For the last term in (3.11), using Lemma 2.10, for σ, θ > 0 such that σ + θ < 1, where we have set α := 2 − σ − θ.This conclude the estimate of the last term of (3.9).
In order to estimate the second term in (3.8) we observe that since u and v are both solutions to the halfwave equation (1.1) we have Consequently, we split the estimate for the second term in (3.8) The first term in (3.12) can be estimated with Hölder and Sobolev inequality, Lemma 2.2, Lemma 3.8.For d ≥ 3 we have Proof.We have We can conclude by Sobolev inequality, Lemma 2.2, observing also that |v| = 1.
For the last term from (3.12) we establish what can be interpreted as a fractional version and extension of the trick (3.4) from [19]. Consequently, In order to estimate |∇| σ (v • |∇|(u − v)) we use Lemma 2.13.We can conclude.
The last term from (3.12) is estimated in the following Lemma 3.10.
We combine this observation with the estimate of Lemma 3.9 for σ = 0 and conclude ˆRd We can conclude.
By now we have estimated the second term in (3.8), which in turn concludes the desired estimate for the second term on the right-hand side of (3.7).
The last term we need to understand is the last term of (3.7), namely we are interested in an estimate for Using again the formula We estimate the first term in (3.13).
We estimate the second term on the right-hand side of (3.13) Lemma 3.12.For d ≥ 2, In particular which readily implies the claim by Hölder's inequality.
We estimate the third term on the right-hand side of (3.13) .
In particular we have .
Proof.We use Lemma 2.5, and have The first term we can estimate with the help of Lemma 3.9, for any small σ ∈ (0, 1 2 ) .
For the second term we can estimate with the help of Lemma 2.11.Taking there α 1 = α 2 = 1+β 2 for a any small β ∈ (0, 1), and From the terms in (3.13) it remains to understand the last one.For this we use the halfwave equation of u and v, (1.1), to write In order to estimate the first term on the right-hand side of (3.14) we first establish the following.Denote with I σ ≡ |∇| −σ the Riesz potential, then we have the following estimate.Observe the power of |∇|v L 2d (R d ) which is crucial here.Lemma 3.14.Let σ ∈ (0, 1), and d ≥ 2, then Proof.The main problem we need to solve is that the term |∇|(u − v) i can not afford any more derivatives.The idea is to factor out |∇| σ derivatives from this term and absorb it into I σ -up to several error terms.We observe first the following algebraic identity, The last term is a double Leibniz type commutator, which we are going to name H|∇|,|∇| σ We apply the above identity to a := v j , b := |∇| 1−σ (u − v) i .Using Sobolev inequality, Lemma 2.2, we find .
The middle term on the right-hand side can be treated further, using that I σ |∇| σ f = f , and using Sobolev inequality, Lemma 2.2, we have . where is the formal adjoint to H |∇| σ (see the estimate of this term below).In summary, we have We treat first and second term in (3.16) at the same time.Namely for α ∈ {0, σ} we discuss .
We observe by Lemma 2.5, for any That is, again using Lemma 2.5, (3.17) If α = 0 the first term in (3.17) is zero.Otherwise we have α = σ and applying twice Leibniz rule estimates, For sufficiently small σ > 0 we can apply Gagliardo-Nirenberg inequality, Lemma 2.3, and obtain This settles the first term in (3.17).
For the second term in (3.17) we estimate if α = 0, In the last line we used Sobolev inequality, Lemma 2.2.If α = σ we adapt this slightly, In the last line we used Gagliardo-Nirenberg inequality, Lemma 2.3.This provides the desired estimate for the second term in (3.17).
For the third term in (3.17) we use Lemma 2.10 (observe that all p i ≥ 2, so (2.6) is trivially satisfied).
If α = 0, we instead use Lemma 2.9, . This provides the desired estimates for the terms in (3.17), i.e. the estimates for first and second term in (3.16).The third term in (3.16), has already been estimated in (3.18).The fourth term in (3.16) we treat by duality.Namely, for some ≤ 1 we have, using also integration by parts, and for some γ < σ, In the second to last step we applied Corollary 2.4 observing that since γ ∈ (0, σ), The last term in (3.16) we estimate via Lemma 2.8, This provides the desired estimates for all terms on the right-hand side of (3.16), and we can conclude.
Lemma 3.14 implies control over the first term on the right-hand side of (3.14) Lemma 3.15.For any α > 0, d ≥ 3 we have .
Proof.For any σ > 0 we may write, using e.g. the Fourier transform to justify this "integration by parts", .
On the other hand, by Lemma 2.13 we have .
Combining the above estimates, we have shown .
The very last term to estimate is the last term on the right-hand side of (3.14) Proof.We can write the term under consideration as a determinant using the wellknown formula a Applying this to a = Consequently, From Lemma 2.14 for any α ∈ ( 1 2 , 1) (since d ≥ 2 there is no further assumption necessary), Recalling that Γ = |∇| (u − v) we conclude.Thus E(0) = 0, and we conclude that E(t 0 ) = 0 for all t 0 ∈ (0, T ).Thus u − v is a constant on R d × [0, T ] -and again since u(0) = v(0) we conclude u ≡ v.