THE CHERN-SIMONS-HIGGS AND THE CHERN-SIMONS-DIRAC EQUATIONS IN FOURIER-LEBESGUE SPACES

. The Chern-Simons-Higgs and the Chern-Simons-Dirac systems in Lorenz gauge are locally well-posed in suitable Fourier-Lebesgue spaces ˆ H s,r . Our aim is to minimize s = s ( r ) in the range 1 < r ≤ 2 . If r → 1 we show that we almost reach the critical regularity dictated by scaling. In the classical case r = 2 the results are due to Huh and Oh. Crucial is the fact that the decisive quadratic nonlinearities fulﬁll a null condition.

The CSD-model was introduced by Cho, Kim and Park [2] and Li-Bhaduri [15]. The equations are gauge invariant. The most common gauges are the Coulomb gauge ∂ j A j = 0 , the Lorenz gauge ∂ µ A µ = 0 and the temporal gauge A 0 = 0. In this paper we exclusively study the Lorenz gauge.
The Fourier-Lebesgue spaces H s,r are defined as the completion of S(R 2 ) with respect to the norm f H s,r = ξ s f L r , where 1 r + 1 r = 1, and f denotes the Fourier transform of f .
The aim is to minimize the regularity of the data so that local well-posedness holds. Persistence of higher regularity is then a consequence of the fact that the results are obtained by a Picard iteration.
Our results for both systems turn out to be almost optimal with respect to scaling for r close to 1 (as shown at the end of this section).
For the CSD system the main result is the following.
satisfy the constraint equation Then there exists T > 0 , T = T ( a µ H s,r , φ H s,r ) such that the CSD system (4),(5),(6) under the Lorenz gauge ∂ µ A µ = 0 has a unique solution Remark. As these results are obtained by a Picard iteration it is well-known that continuous dependence on the data and persistence of higher regularity hold.
In the case r = 2 , s > 1 4 , b = 3 4 + these results were obtained by Huh and Oh [13] for the CSH and the CSD system in Lorenz gauge. Their article is the bases of the present paper. They lowered down the regularity assumptions on the data improving earlier local well posedness results for the CSD system by Huh [12] who had to assume a µ ∈ H 1 2 , ψ 0 ∈ H 5 8 in Lorenz gauge, and a µ ∈ L 2 , ψ 0 ∈ H 1 2 + in Coulomb gauge. In Coulomb gauge local well-posedness was shown for ψ 0 ∈ H 1 4 + by Bournaveas-Candy-Machihara [1] and by a different method by the author [16], who also proved local well-posedness in temporal gauge for ψ 0 ∈ H 3 8 + . In the CSH case Selberg and Tesfahun [19] obtained global well-posedness in Lorenz gauge under a sign condition on V for data a µ ∈Ḣ 1 2 , φ ∈ H 1 , ∂ t φ ∈ L 2 , which is the energy regularity. They also obtained a local solution In temporal gauge global well-posedness in energy space and above was shown by the author [17].
Most of these results used the fact that some or all quadratic nonlinear terms satisfy null conditions, which is also crucial for our paper. We also rely on bilinear estimates in X s,b -spaces by Foschi and Klainerman [5] and d'Ancona-Foschi-Selberg [3].
Well-posedness problems in Fourier-Lebesgue spacesĤ s,r were first considered by Vargas-Vega [20] for 1D Schrödinger equations. Grünrock showed LWP for the modified KdV equation [8], a result which was improved by Grünrock and Vega [10]. Grünrock treated derivative nonlinear wave equations in 3+1 dimensions [9] and obtained an almost optimal result as r → 1 with respect to scaling using calculations by Foschi and Klainerman [5]. Later Grigoryan-Tanguay [7] gave similar results in 2+1 dimensions based on bilinear estimates by Selberg [18].
Systems of nonlinear wave equations in the 2+1 dimensional case for nonlinearities which fulfill a null condition were considered by Grigoryan-Nahmod [6]. This paper is fundamental for the present paper, because the CSH as well as the CSD equations in Lorenz gauge are systems of a similar form.
The paper is organized as follows. In Chapter 2 we start by formulating the general local well-posedness (LWP) theorem for systems of nonlinear wave equations, as it was given by Grünrock [8]. Then estimates for the product of two solutions of the linear wave equation in Fourier-Lebesgue spaces are given, based mainly on Foschi-Klainerman [5], who treated the L 2 -based case. Moreover we also rely on bilinear estimates by d'Ancona-Foschi-Selberg [3] in the L 2 -case and its generalization to the general case by Grigoryan-Tanguay [7]. In addition, we have to estimate cubic nonlinearities. In Chapter 3 the proof of LWP for the Chern-Simons-Higgs system is given, following Huh-Oh [13], who considered the L 2 -case, and proved that the system can be reformulated as a system of nonlinear wave equations, which fulfill a null condition either directly or by duality. For the necessary estimates we rely on the bilinear estimates given in Chapter 2. In Chapter 4 the LWP result for the Chern-Simons-Dirac system is proven. We use the results by d'Ancona-Foschi-Selberg [4] for the case of the Dirac-Klein-Gordon equations concerning the Dirac part, and Huh-Oh [13] again when reformulating the system as a system of nonlinear wave equations with nonlinearities which fulfill a null condition. The necessary estimates which were given by Huh-Oh for the L 2 -case are then implied by the generalizations in Chapter 2.
The CSH system is invariant under the scaling . Under this scaling the norms of the initial data satisfy . Therefore the scaling critical exponent is s c = 2 r − 1 for φ and A µ . For r = 2 we have s c = 0, so that the result of Huh-Oh [13] is 1/4 away from it, whereas for r = 1+ we have s c = 1−, so that for r close to 1 Theorem 1.1 is optimal up to the endpoint.
The CSD system is invariant under the scaling .
The critical exponent is s c = 2 r − 1 for φ and A µ . Similar as for the CSH system we have the result of Huh-Oh in the case r = 2, which is 1/4 away from the critical regularity and the almost optimal result for r = 1+ in Theorem 1.2.
2. Bilinear estimates. We start by collecting some fundamental properties of the solution spaces. We rely on [8]. The spaces X r s,b,± with norm

CHERN-SIMONS-HIGGS AND CHERN-SIMONS-DIRAC 4879
We denote X 2 s,b by H s,b . We also define The "transfer principle" in the following proposition, which is well-known in the case r = 2, also holds for general 1 < r < ∞ (cf. [6], for all combinations of signs ± 1 , ± 2 , then for b > 1 r the following estimate holds: The general local well-posedness theorem is the following (cf. [8], Theorem 1).
are valid. Then there exist T = T ( u 0 Ĥs,r ) > 0 and a unique solution u ∈ X r s,b,± [0, T ] of the Cauchy problem where D is the operator with Fourier symbol |ξ|. This solution is persistent and the mapping data upon solution u 0 → u ,Ĥ s,r → X r s,b,± [0, T 0 ] is locally Lipschitz continuous for any T 0 < T .
The following proposition relies on estimates given by Foschi-Klainerman [5].
. Then the following estimate holds for b > 1 r : Proof. This is a consequence of the transfer principle Prop. 2.1.
In the following we repeatedly use the following bilinear estimates in wave-Sobolev spaces H s,b , which were proven by d'Ancona, Foschi and Selberg in the two-dimensional case n = 2 in [3] in a more general form which include many limit cases which we do not need.
2r . Then the following estimate holds uv X r Proof. Selberg [18] proved this in the case r = 2 . The general case 1 < r ≤ 2 was given by Grigoryan-Tanguay [7], Prop. 3.1, but in fact the case r = 1 is also admissible. More precisely the result follows from [7] after summation over dyadic pieces in a standard way.
As a consequence of these results we obtain Proposition 2.6. Under the assumptions of Prop. 2.4 the following estimate holds:

HARTMUT PECHER
3. The Chern-Simons-Higgs system (Proof of Theorem 1.1). Conveniently we assume κ = 1 . In the Lorenz gauge ∂ µ A µ = 0 the Chern-Simons-Higgs system (1),(2) is equivalent to the following system The initial data are given by Using [13] we may reformulate the CSH system as The homogeneous parts are given by with implicit constants independent of T , where we used the well-known estimates (cf. e.g. [8]) e ±(−i)tD φ 0 X r The Riesz transform is given by R j ± := ∓i −1 D −1 ∂ j and R 0 ± := −1 . Using D λ = ∂ λ − iA λ and the multilinear character of the nonlinearities the estimates concerning (8) are by Theorem 2.1 reduced to and (10) is equivalent to the estimate , which contains the null form Q λµ between φ 2 and φ 0 , so that by duality we have to prove for s > 3 2r − 1 2 and b = 1 The symbol of Q λµ is estimated using [5], Lemma 13.2.
in the case of equal signs, and in the case of unequal signs.
Next we consider the quadratic term in (9). We want to show We The last term is easily treated, because |ξ − ξ | ≤ 1 , so that in fact we only have to prove in this case which is obviously true, if (20) below holds. We now use the Hodge decomposition A i = A df i +A cf i , where the divergence-free part A df and the curl-free part A cf are given by It is well-known (cf. [19] or [13]) that The null form Q 12 was defined before and Q 0 is defined by What remains to be proven are the following estimates: and As before we handle (18) by [5], Lemma 13.2 which reduces matters to We apply Prop. 2.6 with parameters α 0 = s , 2r by our assumption s > 3 2r − 1 2 , and γ = α 1 − 1 r . We also remark that γ > b − 1 2 + under our assumption b = 1 2 + 1 2r + . This implies (20). In order to prove (19) we use Lemma 2.2 and the fact that the symbol of Q 0
Next we prove the estimates for the cubic terms starting with (11). We obtain where we used s + 1 2 > 3 2r and b + s + 1 2 − 1 r > 1 2 + 1 r > 3 2r , so that Prop. 2.5 applies for the first estimate, and the fractional Leibniz rule combined with (21) for the second estimate. Moreover similarly we obtain The first estimate follows as above and the second estimate by Prop. 2.6 with . Consequently by the fractional Leibniz rule we obtain , which implies (11).
Next we have to consider the cubic term A µ A µ φ in equation (9), which requires the estimate First we consider the case r = 1+ , b > 1 r , s > 1 . By the fractional Leibniz rule we reduce to and We obtain by Prop. 2.5 for the first step and Prop. 2.6 for the second step with parameters In a similar way we obtain by Prop. 2.5 and Prop. 2.6 with α 0 = 1 2 , α 1 = s, α 2 = 1 and γ = s + 1 2 − 1 r : , which implies (28) and also (26) in the case r = 1+ . In the case r = 2 , s = 1 4 + , b = 3 4 + we use Prop. 2.3 and obtain By trilinear interpolation between the cases r = 1+ and r = 2 this implies (26) for 1 < r ≤ 2 .
By multilinear interpolation we obtain the desired estimate for 1 < r ≤ 2 .