ALMOST OPTIMAL LOCAL WELL-POSEDNESS FOR THE MAXWELL-KLEIN-GORDON SYSTEM WITH DATA IN FOURIER-LEBESGUE SPACES

. We prove a low regularity local well-posedness result for the Max-well-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces (cid:98) H s,r , where (cid:107) f (cid:107) (cid:98) H s,r = (cid:107)(cid:104) ξ (cid:105) s (cid:98) f ( ξ ) (cid:107) (cid:98) L r (cid:48) , 1 r + 1 r (cid:48) = 1 . The assumed reg- ularity for the data is almost optimal with respect to scaling as r → 1 . This closes the gap between what is known in the case r = 2 , namely s > 34 , and the critical value s c = 12 with respect to scaling.

The Maxwell-Klein-Gordon system describes the motion of a spin 0 particle with mass m self-interacting with an electromagnetic field.
We consider the Cauchy problem in three space dimensions with data φ(x, 0) = φ 0 (x) , ∂ t φ(x, 0) = φ 1 (x), F µν (x, 0) = F 0 µν (x) . The potential A is not uniquely determined but one has gauge freedom. The Maxwell-Klein-Gordon equation is namely invariant under the gauge transformation φ → φ = e iχ φ , A µ → A µ = A µ + 3304 HARTMUT PECHER ∂ µ χ for any χ : R n+1 → R. We exclusively consider the Lorenz gauge ∂ µ A µ = 0. However we remark that even this gauge does not uniquely determine A , because any χ satisfying χ = 0 preserves the Lorenz gauge condition, so that one has to add a further condition (cf. (1.15) below) in order to obtain a unique potential.
We can reformulate the system (1.1), (1.2) under the Lorenz condition as follows: thus (using the notation ∂ = (∂ 0 , ∂ 1 , ..., ∂ n )): and Conversely, if A µ = −j µ and F µν := ∂ µ A ν − ∂ ν A µ and the Lorenz condition (1.6) holds then where f H s,r = ξ s f (ξ) L r , 1 r + 1 r = 1 . The system (1.7), (1.8) is invariant under the scaling A λ (t, x) = λA(λt, λx) , φ λ (t, x) = λφ(λt, λx) , i.e. A λ , φ λ are solutions, if A, φ are. The homogeneous Fourier-Lebesgue norm of the initial data scales like , so that the critical exponent with respect to scaling is s c = 3 r −1 for A µ as well as φ, thus the critical exponent for No well-posedness is expected for s < s c . Thus, up to the endpoint the best one can hope for is local well-posedness for s > s c .
In the classical case r = 2 and Coulomb gauge ∂ j A j = 0 Klainerman and Machedon [11] showed global well-posedness in energy space and above, i.e. for data φ 0 ∈ H s , φ 1 ∈ H s−1 , a 0ν ∈ H s ,ȧ 0ν ∈ H s−1 with s ≥ 1 in n = 3 dimensions. They made the fundamental observation that the nonlinearities fulfill a null condition. This global well-posedness result was improved by Keel, Roy and Tao [10], who had only to assume s > dimensions and in Coulomb gauge. Machedon and Sterbenz [13] proved local wellposedness even in the almost critical range s > 1 2 , but had to assume a smallness assumption on the data.
In Lorenz gauge ∂ µ A µ = 0 and data in Sobolev spaces H s , which was considered much less in the literature, because the nonlinear term Im(φ∂ µ φ) has no null structure, and three space dimensions the most important progress was made by Selberg and Tesfahun [18] who were able to circumvent the problem of the missing null condition in the equations for A µ by showing that the decisive nonlinearities in the equations for φ as well as F µν fulfill such a null condition which allows to show that global well-posedness holds for large, finite energy data s = 1 . The potential possibly loses some regularity compared to the data but as remarked also by the authors this is not the main point because one is primarily interested in the regularity of φ and F µν , which both preserve the regularity. Below energy level the author [16] was also able to show local well-posedness for large data in this sense, provided s > 3 4 . This means that for large data there is still a gap of 1 4 to s c > 1 2 predicted by scaling both in Coulomb and Lorenz gauge.
We work in Lorenz gauge and close this gap in the limit r → 1 for large data in Fourier-Lebesgue spaces H s,r for 1 < r ≤ 2 , thus leaving the H s -scale of data (r = 2). This is remarkable in view of the fact that one of the nonlinearities does not fulfill a null condition. In some sense it is the first large data almost optimal local well-posedness result for the Maxwell-Klein-Gordon equations no matter which gauge is considered.
A null structure in Lorenz gauge was first detected for the Maxwell-Dirac system by d'Ancona, Foschi and Selberg [1].
In two space dimensions in Coulomb gauge Czubak and Pikula [4] proved local well-posedness provided that φ 0 ∈ H s , φ 1 ∈ H s−1 , a 0ν ∈ H r ,ȧ 0ν ∈ H r−1 , where 1 ≥ s = r > 1 2 or s = 5 8 + , r = 1 4 + . In four space dimensions Selberg [17] showed local well-posedness in Coulomb gauge for s > 1 , which is almost critical. In Lorenz gauge the author [14] considered also the case n ≥ 4 and proved local well-posedness for s > n 2 − 5 6 . In order to achieve an almost optimal local well-posedness result for n = 3 we consider the Lorenz gauge and Cauchy data in Fourier-Lebesgue spaces H s,r for 1 < r ≤ 2 . Data in spaces of this type were previously considered by Grünrock [7] and Grünrock-Vega [9] for KdV and modified KdV equations. Grünrock [8] used these spaces in order to prove almost optimal low regularity local well-posedness also for wave equations with quadratic derivative nonlinearities for n = 3. For wave equations with a nonlinearity which fulfills a null condition this was also shown in the case n = 2 by Grigoryan-Nahmod [6]. These results relied on a modification of bilinear estimates which were given by Foschi-Klainerman [5] in the classical L 2 -case.

HARTMUT PECHER
We show that for (admissible) data φ(0) ∈ H s,r , (∂ t φ)(0) ∈ H s−1,r and (∇A µ ) (0) ∈ H s−1,r , (∂ t A µ )(0) ∈ H s−1,r we obtain a solution of (1.7),(1.8) , where φ belongs to X r s,b, r and s > 5 2r − 1 2 , so that s → 2 as r → 1 , which is almost optimal with respect to scaling. We also obtain ∇A µ ∈ X r l,b,+ [0, T ] + X r l,b,+ [0, T ] , where l > 2 r − 1, so that l → 2 as r → 1 , but l < s due to the missing null condition in the term Im(φ∂φ). However this is of minor interest, because the really important fact is that F µν ∈ X r s−1,b, r . This is a consequence of the fact that F µν fulfills a wave equation with null forms in the quadratic inhomogeneous terms (see (1.29),(1.30) below).
Fundamental are of course the bi-and trilinear estimates for the nonlinearities where the quadratic terms have null structure except one term, namely Im(φ∂ µ φ). This is the reason why A µ (possibly) loses regularity in time. We rely on the bilinear estimates of Foschi-Klainerman [5], which were already successfully applied by Grünrock [8] and Grigoryan-Nahmod [6]. The general local well-posedness theorem for nonlinear systems of wave equations (and also other types of evolution equations) in X r s,b -spaces , which reduces the problem to multilinear estimates for the nonlinear terms, goes back to Grünrock [7], cf. also [6]. For the Cauchy problem for the Maxwell-Klein-Gordon system in Lorenz gauge with L 2 -based data we rely on the author's paper [15], which is a refinement of the earlier paper [16]. We use the following notation. Let f denote the Fourier transform of f with respect to space and time as well as with respect to space, which should be clear from the context. We define the wave-Sobolev spaces X r s,b,± for 1 < r ≤ 2 and 1 r + 1 r = 1 as the completion of the Schwarz space S(R n+1 ) with respect to the norm u X r s,b,± = ξ s τ ± |ξ| b u(τ, ξ) L r τ ξ and X r s,b,± [0, T ] as the space of the restrictions to [0, T ] × R n . We also define the spaces X r s,b as the completion of S(R n+1 ) with respect to the norm t + ∆ is the d'Alembert operator. a± = a ± for a sufficiently small > 0 and a + + = (a+)+ . Next we formulate our main results. We assume the Lorenz condition and Cauchy data 14) which are assumed to fulfill a 00 =ȧ 00 = 0 , (1.15) and the following compatibility conditions (1.15) may be assumed because otherwise the Lorenz condition does not determine the potential uniquely. As remarked already any function χ in the gauge transformation with χ = 0 preserves the Lorenz condition. Thus in order to obtain uniqueness we assume that χ moreover fulfills ∆χ(0) = −∂ j a 0j and (∂ t χ)(0) = −a 00 . This implies by the gauge transformation a 00 = a 00 jk . These conditions imply the following regularity for the initial data ∇a 0j ∈ H s−1,r ,ȧ 0j ∈ H s−1,r . (1.20) We prefer to rewrite our system (1.7),(1.8) as a first order (in t) system. Let We obtain the equivalent system The initial data are given by We split A ± = A hom ± + A inh ± into its homogeneous and inhomogeneous part, where (i∂ t ± D)A hom ± = 0 with data as in (1.27) and (1.28) and A inh ± is the solution of (1.22) with zero data.
Our first main theorem reads as follows: r and 0 is a small positive number. Remark: The solution depends continuously on the initial data and persistence of higher regularity holds (see Theorem 1.4 below).
In order to obtain the optimal regularity for F µν it is possible to derive from Maxwell's equations (1.1) and (1.5) the following wave equations, where we refer to [17], section 3.2 or [15], section 2.
We prove as a consequence of this system or its equivalent first order (in t) system the second main result.
where 0 is a small positive number.
For the following basic properties of X r s,b -spaces as well as a general local wellposedness theorem for nonlinear systems we refer to Grünrock [7].
For general phase functions φ : Then these spaces have the following properties: The following general local well-posedness theorem is an obvious generalization of [7], Thm. 1.
Here ω and ω 1 are increasing smooth functions. Then the Cauchy problem This solution depends locally lipschitzian on the data and higher regularity is preserved.
Remark: This theorem can be generalized to systems of equations in a straightforward manner, especially to the Maxwell-Klein-Gordon system in the form (1.21), (1.22).
The following estimate applies: Proof. By [5], Lemma 13.2 we obtain Elliptic part: We use the first bound for the elliptic part, we have τ = |η| + |ξ − η| so that τ ≥ |ξ| , and we have to show
Then the following estimate applies
We want to apply this result to the nonlinearity A µ ∂ µ φ and recall the known null structure of this term, which can be found in [18] or [15]. We use the Hodge decomposition A = A cf +A df , where A cf = ∆ −1 ∇(∇·A) and A df = −∆ −1 ∇×∇×A are the curl-free and divergence-free part, respectively. This decomposes the term as follows: Using the Lorenz gauge ∂ t A 0 = ∇ · A we obtain A cf = ∆ −1 ∇∂ t A 0 , thus this reads as follows: Thus the symbol of P 1 behaves as follows (cf. [17]): These bounds for the symbol of A µ ∂ µ φ are now combined with Lemma 2.5 and Lemma 2.4, which implies the following result.
Lemma 2.6. For r = 1+ the following estimate applies: Next we want to estimate the term Im(φ∂φ) .

This implies
, where we applied Lemma 2.8 in the last step. The second estimate is proven in the same way.
Lemma 2.11. For r = 1+ and b > 1 r the following estimates are true: ∇(A|φ| 2 ) X r 0,0 for 0 < < 3 2 (1 − 1 r ) . Proof. The first estimate is proven exactly like Lemma 2.9. For the last estimate we modify the proof of Lemma 2.7 for the case α 0 = 1 in order to prove: We simply replace |ξ| α0r = |ξ| r by τ r and obtain the same estimate for I in the elliptic case, whereas in the hyperbolic case we use |τ | ≤ |ξ| , which leads to the previous bounds for I. Using this new estimate we may handle the term ∂ t (A|φ| 2 ) X r 0,0 exactly like A|φ| 2 Ẋr 1,0 in Lemma 2.9. Now we interpolate the bi-and trilinear estimates in X r s,b -spaces for r = 1+ just proven and for r = 2, where for the latter we rely on the results in [15].