\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 174, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/174\hfil Chern-Simons-Dirac systems] {Low regularity solutions for Chern-Simons-Dirac systems in the temporal and Coulomb gauge} \author[H. Pecher \hfil EJDE-2016/174\hfilneg] {Hartmut Pecher} \address{Hartmut Pecher \newline Fakult\"at f\"ur Mathematik und Naturwissenschaften, Bergische Universit\"at Wuppertal, Gau{\ss}str. 20, 42119 Wuppertal, Germany} \email{pecher@math.uni-wuppertal.de} \thanks{Submitted November 10, 2015. Published July 6, 2016.} \subjclass[2010]{35Q40, 35L70} \keywords{ Chern-Simons-Dirac;local well-posedness; Coulomb gauge; \hfill\break\indent temporal gauge} \begin{abstract} We prove low regularity local well-posedness results in Bourgain-Klainerman-Machedon spaces for the Chern-Simons-Dirac system in the temporal gauge and the Coulomb gauge. Under slightly stronger assumptions on the data we also obtain ``unconditional'' uniqueness in the natural solution spaces. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction and statement of main results} Consider the Chern-Simons-Dirac system in two space dimensions \begin{gather}\label{1} i \partial_t \psi + i \alpha^j \partial_j \psi = m \beta \psi - \alpha^{\mu} A_{\mu} \psi\\ \label{2} \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} = -2 \epsilon_{\mu \nu \lambda} \langle \psi , \alpha^{\lambda} \psi \rangle \end{gather} with initial data \begin{equation} \label{2'} \psi(0) = \psi_0, \quad A_{\mu}(0) = a_{\mu} \, , \end{equation} where we use the convention that repeated upper and lower indices are summed, Latin indices run over 1,2 and Greek indices over 0,1,2 with Minkowski metric of signature ($+$,$-$,$-$). Here $\psi : \mathbb{R}^{1+2} \to \mathbb{C}^2$, $A_{\nu} : \mathbb{R}^{1+2} \to \mathbb{R}$, $m \in \mathbb{R}$. $\alpha^1,\alpha^2, \beta$ are hermitian ($ 2 \times 2$)-matrices satisfying $\beta^2 =(\alpha^1)^2 = (\alpha^2)^2 = I $, $ \alpha^j \beta + \beta \alpha^j = 0$, $\alpha^j \alpha^k + \alpha^k \alpha^j = 2 \delta^{jk} I $, $ \alpha^0 = I $. $\langle \cdot,\cdot \rangle $ denotes the $\mathbb{C}^2$-scalar product. A particular representation is given by \[ \alpha^1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \alpha^2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \beta = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \] $\epsilon_{\mu \nu \lambda}$ is the totally skew-symmetric tensor with $\epsilon_{012} = 1$ . This model was proposed by Cho, Kim and Park \cite{CKP}, and by Li and Bhaduri \cite{LB}. The equations are invariant under the gauge transformations $$ A_{\mu} \to A'_{\mu} = A_{\mu} + \partial_{\mu} \chi \, , \quad \psi \to \psi' = e^{i\chi} \phi \, . $$ The most common gauges are the Coulomb gauge $\partial^j A_j =0$, the Lorenz gauge $\partial^{\mu} A_{\mu} = 0$ and the temporal gauge $A_0 = 0$. Local well-posedness for data with minimal regularity assumptions was shown by Huh \cite{H} in the Lorenz gauge for data $\psi_0 \in H^{\frac{5}{8}} $, $a_{\mu} \in H^{1/2}$ using a null structure, in the Coulomb gauge for $\psi_0 \in H^{\frac{1}{2}+\epsilon}$, $ a_i \in L^2$, and in temporal gauge for $\psi_0 \in H^{\frac{3}{4}+\epsilon}$, $a_j \in H^{\frac{3}{4}+\epsilon} + L^2$, both without using a null structure. The result in Lorenz gauge was improved by Huh-Oh \cite{HO} where the regularity of the data was lowered down to $\psi_0 \in H^s$, $a_{\mu} \in H^s$ with $s >1/4$. Their proof relies also on a null structure in the nonlinear terms of the Dirac equation as well as the wave equation. They apply a Picard iteration in Bourgain-Klainerman-Machedon spaces $X^{s,b}$, which implies uniqueness in these spaces. Independently Okamoto \cite{O} proved a similar result in Lorenz as well as Coulomb gauge also using a null structure of the system. The methods of Okamoto and Huh-Oh are different. Okamoto reduces the problem to a single Dirac equation with cubic nonlinearity for $\psi$, which does not contain $A_{\mu}$ any longer. From a solution $\psi$ of this equation the potentials $A_{\mu}$ can be constructed by solving a wave equation in Lorenz gauge and an elliptic equation in Coulomb gauge. Huh-Oh on the other hand directly solve a coupled system of a Dirac equation for $\psi$ and a wave equation for $A_{\mu}$. Recently Bournaveas-Candy-Machihara \cite{BCM} proved local well-posedness in Coulomb gauge under similar regularity assumptions without use of a null structure. Their proof relies on a bilinear Strichartz estimate given by Klainerman-Tataru \cite{KT}. A low regularity local well-posedness result in temporal gauge was given by Tao \cite{T1} for the Yang-Mills equations. In the present paper we consider the temporal gauge as well as the Coulomb gauge. In temporal gauge we improve the result of Huh \cite{H} to data $\psi_0 \in H^s$, $a_j \in H^{s+\frac{1}{8}}$ with $s >3/8$. We use Bourgain-Klainerman-Machedon spaces $X^{s,b}$ adapted to the phase functions $\tau \pm |\xi|$ on one hand and $\tau$ on the other hand. We decompose $A_j$ into its divergence-free part $A^{df}_j$ and its curl-free part $A^{cf}_j$. The main problem here is that there seems to be no null structure in the nonlinearity $A^{cf}_j \alpha^j \psi$ in the Dirac equation whereas in Lorenz gauge $A^{cf}_{\mu} \alpha^{\mu} \psi$ has such a null structure. In fact all the other terms possess such a null structure. However we are not able to use it for an improvement of our result. We apply the bilinear estimates in wave-Sobolev spaces established in d'Ancona-Foschi-Selberg \cite{AFS} which rely on Strichartz estimates. Morover we use a variant of an estimate for the $L^6_x L^2_t$ - norm for the solution of the wave equation which goes back to Tataru and Tao. When applying this estimate we partly follow Tao's arguments in the case of the Yang-Mills equations \cite{T1}. We prove existence and uniqueness in $X^{s,b}$ - spaces first (Theorem \ref{Theorem 1.1}). Then we prove unconditional uniqueness under the stronger assumption $s > \frac{19}{40}$ (Theorem \ref{Theorem 1.2}) by using an idea of Zhou \cite{Z}. In Coulomb gauge we make the same regularity assumptions as Okamoto \cite{O} and Bournaveas-Candy-Machihara \cite{BCM}, namely $\psi_0 \in H^{\frac{1}{4}+\epsilon}$, and also reduce the problem to a single Dirac equation with cubic nonlinearity. We give a short (alternative) proof of local well-posedness in $X^{s,b}$ - spaces without use of a null structure (Theorem \ref{Theorem 1.1'}) using d'Ancona-Foschi-Selberg \cite{AFS} (cf. Proposition \ref{Prop. 1.1}). We also prove unconditional uniqueness in the space $\psi \in C^0([0,T],H^s)$ under the assumption $s > 1/3$ (Theorem \ref{Theorem 1.2'}). We first give some notation. We denote the Fourier transform with respect to space and time by $\,\widehat{}$ . The operator $|\nabla|^{\alpha}$ is defined by $\mathcal{F}(|\nabla|^{\alpha} f)(\xi) = |\xi|^{\alpha} (\mathcal{F}f)(\xi)$, where $\mathcal{F}$ is the Fourier transform, and similarly $ \langle \nabla \rangle^{\alpha}$, where $\langle \cdot \rangle := (1+|\cdot|^2)^{1/2}$. The inhomogeneous and homogeneous Sobolev spaces are denoted by $H^{s,p}$ and $\dot{H}^{s,p}$, respectively. For $p=2$ we simply denote them by $H^s$ and $\dot{H}^s$. We repeatedly use the Sobolev embeddings $\dot{H}^{s,p} \hookrightarrow L^q$ for $10$ , so that $a 0$ and $s > 3/8$. The Chern-Simons-Dirac system \eqref{1}, \eqref{2}, \eqref{2'} in temporal gauge $A_0=0$ with data $\psi_0 \in H^s(\mathbb{R}^2)$, $a_j \in H^{s+\frac{1}{8}}(\mathbb{R}^2)$, satisfying the compatibility condition $\partial_1 a_2- \partial_2 a_1 = -2 \langle \psi_0, \psi_0 \rangle$, has a local solution $$ \psi \in C^0([0,T],H^s(\mathbb{R}^2)) , \quad |\nabla|^{\epsilon} A_j \in C^0([0,T],H^{s+\frac{1}{8}-\epsilon}(\mathbb{R}^2)) \, . $$ More precisely $\psi = \psi_+ + \psi_-$ with $\psi_{\pm} \in X^{s,\frac{1}{2}+}_{\pm}[0,T]$. If $ A = A^{df} + A^{cf} $ is the decomposition into its divergence-free part and its ``curl-free'' part, where \begin{gather*} A^{df} = (-\Delta)^{-1}(\partial_2(\partial_1A_2-\partial_2 A_1), \partial_1(\partial_2 A_1-\partial_1 A_2)) \, , \\ A^{cf} = -(-\Delta)^{-1}(\partial_1(\partial_1 A_1 + \partial_2 A_2),\partial_2(\partial_1 A_1 + \partial_2 A_2)) = -(-\Delta)^{-1} \nabla \operatorname{div}A \, , \end{gather*} one has $$ A^{cf} \in X^{s+\frac{1}{8},\frac{1}{2}+}_{\tau =0} [0,T] , \quad |\nabla|^{\epsilon} A^{df} \in X^{s+\frac{3}{8}-\epsilon,\frac{1}{2}+}_{|\tau| =|\xi|}[0,T] $$ and in these spaces uniqueness holds. Moreover we have $\psi_{\pm} \in X^{s,1}_{\pm} [0,T]$. \end{theorem} \subsection*{Remark} The Chern-Simons-Dirac system is invariant under the scaling $$ \psi^{(\lambda)}(t,x) = \lambda\psi(\lambda t,\lambda x) \, , \, A^{(\lambda)}(t,x) = \lambda A_{\mu}(\lambda t, \lambda x) \, . $$ Thus in 2+1 dimensions the scaling critical Sobolev exponent is $s=0$, i.e. $\psi_0$, $a_{\mu} \in H^s = L^2$. In Lorenz gauge Huh-Oh \cite{HO} remarked that their result $s > 1/4$ is probably optimal in view of Zhou \cite{Z1}, who proved that is the case for a system of nonlinear wave equations with nonlinearities, which fulfill a null condition. In our case of the temporal gauge however the system is reduced to a coupled system of a wave equation for $\psi$ and a transport equation for $A^{cf}$ where null conditions seem to be not useful because they are only adapted for wave equations. Nevertheless it would be desirable to improve our result to $s > 1/4$ for $\psi_0$ and $a_j$ . \begin{theorem}\label{Theorem 1.2} Let the assumptions of Theorem \ref{Theorem 1.1} be fulfilled. If $s > 19/40$, the solution of \eqref{1}, \eqref{2}, \eqref{2'} is unique in the space $ \psi \in C^0([0,T],H^s(\mathbb{R}^2))$, $A^{cf} \in C^0([0,T],H^{s+\frac{1}{8}}(\mathbb{R}^2))$, $|\nabla|^{\epsilon} A^{df} \in C^0([0,T],H^{s+\frac{3}{8}-\epsilon}(\mathbb{R}^2))$. \end{theorem} Consider now the Coulomb gauge condition $\partial_j A^j = 0$. In this case one easily checks using \eqref{2} that the potentials $A_{\mu}$ satisfy the elliptic equations \begin{equation} \label{1.10} \begin{gathered} A_0 = \Delta^{-1}(\partial_2 \langle \psi,\alpha^1 \psi \rangle - \partial_1 \langle \psi,\alpha^2 \psi \rangle) \, , \\ A_1 = \Delta^{-1} \partial_2 \langle \psi,\psi \rangle \, , \quad A_2 = - \Delta^{-1} \partial_1 \langle \psi,\psi \rangle \, . \end{gathered} \end{equation} Inserting this into \eqref{1} we obtain \begin{equation}\label{1.11} i \partial_t \psi + i \alpha^j \partial_j \psi = m \beta \psi + N(\psi,\psi,\psi) \, , \end{equation} where \begin{align*} &N(\psi_1,\psi_2,\psi_3) \\ &= \Delta^{-1}\left( \partial_1 \langle \psi_1,\alpha_2 \psi_2 \rangle - \partial_2 \langle \psi_1,\alpha_1 \psi_2 \rangle + \partial_2 \langle \psi_1,\psi_2 \rangle \alpha_1 - \partial_1 \langle \psi_1,\psi_2 \rangle \alpha_2 \right) \psi_3 \, . \end{align*} In the sequel we consider this nonlinear Dirac equation with initial condition \begin{equation} \label{1.12} \psi(0) = \psi_0 \, . \end{equation} Using an idea of d'Ancona - Foschi -Selberg \cite{AFS1} we simplify \eqref{1.11} by considering the projections onto the one-dimensional eigenspaces of the operator $-i \alpha \cdot \nabla = -i \alpha^j \partial_j$ belonging to the eigenvalues $ \pm |\xi|$. These projections are given by $\Pi_{\pm} =$ $\Pi_{\pm}(D)$, where $ D =\frac{\nabla}{i} $ and $\Pi_{\pm}(\xi) = \frac{1}{2}(I\pm \frac{\xi}{|\xi|} \cdot \alpha) $. Then $ -i\alpha \cdot \nabla = |D| \Pi_+(D) - |D| \Pi_-(D) $ and $ \Pi_{\pm}(\xi) \beta = \beta \Pi_{\mp}(\xi) $. Defining $ \psi_{\pm} := \Pi_{\pm}(D) \psi$, the Dirac equation can be rewritten as \begin{equation} \label{4'} (-i \partial_t \pm |D|)\psi_{\pm} = m\beta \psi_{\mp} + \Pi_{\pm}N(\psi_+ + \psi_-,\psi_+ + \psi_-, \psi_+ + \psi_-) \, . \end{equation} The initial condition is transformed into \begin{equation} \label{6'} \psi_{\pm}(0) = \Pi_{\pm}\psi_0 \, . \end{equation} We now formulate our results in the case of the Coulomb gauge. \begin{theorem}\label{Theorem 1.1'} Assume $\psi_0 \in H^s(\mathbb{R}^2)$ with $s > 1/4$. Then \eqref{1.11},\eqref{1.12} is locally well-posed in $H^s(\mathbb{R}^2)$. More precisely there exists $T>0$, such that there exists a unique solution $\psi = \psi_+ + \psi_-$ with $\psi_{\pm} \in X^{s,\frac{1}{2}+}_{\pm}[0,T]$. This solution belongs to $C^0([0,T],H^s(\mathbb{R}^2))$. \end{theorem} The unconditional uniqueness result is the following. \begin{theorem}\label{Theorem 1.2'} Assume $ \psi_0 \in H^s(\mathbb{R}^2)$ with $ s > 1/3$. The solution of \eqref{1.11}, \eqref{1.12} is unique in $C^0([0,T],H^s(\mathbb{R}^2))$. \end{theorem} Fundamental for the proof of our theorems are the following bilinear estimates in wave-Sobolev spaces which were proven by d'Ancona, Foschi and Selberg in the two dimensional case $n=2$ in \cite{AFS} in a more general form which include many limit cases which we do not need. \begin{proposition} \label{Prop. 1.1} Let $n=2$. The estimate $$ \|uv\|_{X_{|\tau|=|\xi|}^{-s_0,-b_0}} \lesssim \|u\|_{X^{s_1,b_1}_{|\tau|=|\xi|}} \|v\|_{X^{s_2,b_2}_{|\tau|=|\xi|}} $$ holds, provided the following conditions hold: \begin{gather*} b_0 + b_1 + b_2 > \frac{1}{2},\quad b_0 + b_1 \ge 0, \\ b_0 + b_2 \ge 0, \quad b_1 + b_2 \ge 0, \\ s_0+s_1+s_2 > \frac{3}{2} -(b_0+b_1+b_2), \\ s_0+s_1+s_2 > 1 -\min(b_0+b_1,b_0+b_2,b_1+b_2), \\ s_0+s_1+s_2 > \frac{1}{2} - \min(b_0,b_1,b_2), \quad s_0+s_1+s_2 > \frac{3}{4}, \\ (s_0 + b_0) +2s_1 + 2s_2 > 1, \quad 2s_0+(s_1+b_1)+2s_2 > 1, \\ 2s_0+2s_1+(s_2+b_2) > 1, \quad s_1 + s_2 \ge \max(0,-b_0), \\ s_0 + s_2 \ge \max(0,-b_1), \quad s_0 + s_1 \ge \max(0,-b_2) \, . \end{gather*} \end{proposition} Another decisive tool are the estimates for the wave equation in the following proposition. \begin{proposition} \label{prop1.2} The following estimates hold \begin{align} \label{15} \|u\|_{L^6_{xt}} & \lesssim \|u\|_{X^{\frac{1}{2},\frac{1}{2}+}_{|\tau|=|\xi|}} \, , \\ \label{16} \|u\|_{L^p_x L^2_t} & \lesssim \|u\|_{X^{\frac{1}{2}-\frac{2}{p},\frac{1}{2}+}_{|\tau|=|\xi|}} \quad \mbox{for} \,\, 6 \le p < \infty \, ,\\ \label{16'} \mbox{especially} \quad \|u\|_{L^6_x L^2_t} &\lesssim \|u\|_{X^{\frac{1}{6},\frac{1}{2}+}_{|\tau|=|\xi|}} \, , \\ \label{16''} \|u\|_{L^{\infty}_x L^2_t} &\lesssim \|u\|_{X^{\frac{1}{2}+,\frac{1}{2}+}_{|\tau|=|\xi|}} \, , \\ \label{16'''} \|u\|_{L^{\infty}_x L^{2+}_t} &\lesssim \|u\|_{X^{\frac{1}{2}+,\frac{1}{2}+}_{|\tau|=|\xi|}} \, , \\ \label{18} \|u\|_{L^6_x L^{2+}_t} &\lesssim \|u\|_{X^{\frac{1}{6}+,\frac{1}{2}+}_{|\tau|=|\xi|}} \, , \\ \label{17} \|u\|_{L^4_x L^{2+}_t} &\lesssim \|u\|_{X^{\frac{1}{8}+,\frac{3}{8}+}_{|\tau|=|\xi|}} \, , \\ \label{19} \|u\|_{L^p_x L^{2+}_t} & \lesssim \|u\|_{X^{\frac{1}{2}-\frac{2}{p}+,\frac{1}{2}+}_{|\tau|=|\xi|}} \quad \mbox{for} \,\, 6 \le p < \infty \, . \end{align} \end{proposition} \begin{proof} \eqref{15} is the standard Strichartz estimate combined with the transfer principle. Concerning \eqref{16} we use \cite{KMBT} (appendix by D. Tataru) Thm. B2: $$ \|\mathcal{F}_t u \|_{L^2_{\tau} L^6_x} \lesssim \|u_0\|_{\dot{H}^{\frac{1}{6}}_x} \, , $$ if $u=e^{it |\nabla|}u_0$ , and $\mathcal{F}$ denotes the Fourier transform with respect to time. This implies by Plancherel, Minkowski's inequality and Sobolev's embedding theorem $$ \|u\|_{L^p_x L^2_t} = \|\mathcal{F}_t u\|_{L^p_x L^2_{\tau}} \lesssim \|\mathcal{F}_t u\|_{L^2_{\tau} L^p_x} \lesssim \|\mathcal{F}_t u\|_{L^2_{\tau} H^{\frac{1}{3}-\frac{2}{p},6}_x} \lesssim \|u_0\|_{H^{\frac{1}{2}-\frac{2}{p},2}_x} \, . $$ The transfer principle gives \eqref{16}. \eqref{16''} follows similarly using $H^{\frac{1}{3}+,6}_x \hookrightarrow L^{\infty}_x$. \eqref{18} is obtained by interpolation between \eqref{16'} and \eqref{15}, and \eqref{17} by interpolation between \eqref{18} and the trivial identity $\|u\|_{L^2_{xt}} = \|u\|_{X^{0,0}_{|\tau|=|\xi|}}$. Moreover we obtain \eqref{16'''} and \eqref{19} by interpolation between \eqref{16''} and \eqref{16}, resp., and the estimate $\|u\|_{L^{\infty}_{xt}} \lesssim \|u\|_{X^{1+,\frac{1}{2}+}_{|\tau|=|\xi|}}$ . \end{proof} \section{Reformulation of the problem in temporal gauge} Imposing the temporal gauge condition $A_0 =0$ the system \eqref{1}, \eqref{2} is equivalent to \begin{align} \label{3} &i \partial_t \psi + i \alpha^j \partial_j \psi = m \beta \psi - \alpha^{j} A_{j} \psi\\ \label{4} &\partial_t A_1 = -2 \langle \psi , \alpha^2 \psi \rangle \,, \quad \partial_t A_2 = 2 \langle \psi , \alpha^1 \psi \rangle \\ \label{5} &\partial_1 A_2 - \partial_2 A_1 = -2 \langle \psi, \psi \rangle \, . \end{align} We first show that \eqref{5} is fulfilled for any solution of \eqref{3}, \eqref{4}, if it holds initially, i.e., if the following compatability condition holds: \begin{equation} \partial_1 A_2(0) - \partial_2 A_1(0) = -2 \langle \psi(0),\psi(0) \rangle \, , \end{equation} which we assume from now on. Indeed one easily calculates using \eqref{3}: \begin{equation} \label{8} \partial_t \langle \psi,\psi \rangle = -\partial_j \langle \psi, \alpha^j \psi \rangle \, , \end{equation} which implies by \eqref{4} $$ \partial_t(\partial_1 A_2 - \partial_2 A_1) = 2 \partial_j \langle \psi,\alpha^j \psi \rangle = -2 \partial_t \langle \psi, \psi \rangle \, , $$ so that \eqref{5} holds, if it holds initially. Thus we only have to solve \eqref{3} and \eqref{4}. We decompose $A=(A_1,A_2)$ into its divergence-free part $A^{df}$ and its ``curl-free'' part $A^{cf}$, namely $A=A^{df} + A^{cf}$, where \begin{gather*} A^{df} = (-\Delta)^{-1}(\partial_2(\partial_1A_2-\partial_2 A_1), \partial_1(\partial_2 A_1-\partial_1 A_2)) \, , \\ A^{cf} = -(-\Delta)^{-1}(\partial_1(\partial_1 A_1 + \partial_2 A_2), \partial_2(\partial_1 A_1 + \partial_2 A_2)) = -(-\Delta)^{-1} \nabla \operatorname{div}A \, . \end{gather*} Then \eqref{5} and \eqref{4} imply \begin{gather} \label{6} A^{df} = -2(-\Delta)^{-1} (\partial_2 \langle \psi,\psi \rangle, -\partial_1 \langle \psi, \psi \rangle) \,, \\ \label{7} \partial_t A_j^{cf} = -2(-\Delta)^{-1} \partial_j(\partial_2 \langle \psi,\alpha^1 \psi \rangle - \partial_1 \langle \psi,\alpha^2 \psi \rangle) \, . \end{gather} Reversely, defining $A=A^{df}+A^{cf}$, we show that our new system \eqref{3}, \eqref{6}, \eqref{7} implies \eqref{3}, \eqref{4}, \eqref{5}, so that both systems are equivalent. It only remains to show that \eqref{4} holds. By \eqref{6}, \eqref{7}, \eqref{8} we obtain \begin{align*} \partial_t A_1 & = \partial_t A_1^{df} + \partial_t A_1^{cf} \\ & = -2(-\Delta)^{-1}\big( \partial_2 \partial_t \langle \psi,\psi \rangle + \partial_1(\partial_2 \langle \psi,\alpha^1 \psi \rangle - \partial_1 \langle \psi, \alpha^2 \psi \rangle ) \big) \\ & = 2(-\Delta)^{-1}\big( \partial_2 \partial_j \langle \psi,\alpha^j \psi \rangle - \partial_1(\partial_2 \langle \psi,\alpha^1 \psi \rangle - \partial_1 \langle \psi, \alpha^2 \psi \rangle ) \big) \\ & = 2(-\Delta)^{-1}(\partial_2^2 + \partial_1^2) \langle \psi,\alpha^2 \psi \rangle = -2 \langle \psi, \alpha^2 \psi \rangle \end{align*} and similarly $$ \partial_t A_2 = 2 \langle \psi,\alpha^1 \psi \rangle \, . $$ In the same way in which we obtained \eqref{4'} the Dirac equation \eqref{3} can be rewritten as \begin{equation}\label{9} (- i\partial_t \pm|\nabla|)\psi_{\pm} = -m\beta \psi_{\mp} - \Pi_{\pm}(\alpha^j A_j \psi) \, , \end{equation} where $A_j = A_j^{df} + A_j^{cf}$, and in \eqref{6},\eqref{7} and \eqref{9} we replace $\psi$ by $\psi_+ + \psi_-$. \section{Proof of Theorem \ref{Theorem 1.1}} Taking the considerations of the previous section into account Theorem \ref{Theorem 1.1} reduces to the following proposition and its corollary. \begin{proposition} \label{Prop. 2.1} Let $\epsilon >0$ and $s > 3/8$. Then there exists $T>0$ such that system \eqref{6}, \eqref{7}, \eqref{9} has a unique local solution $\psi_{\pm} \in X^{s,\frac{1}{2}+}_{\pm}[0,T] \, , \, A^{cf} \in X^{s+\frac{1}{8},\frac{1}{2}+}_{\tau =0} [0,T] $. Also $A^{df}$ satisfies $|\nabla|^{\epsilon} A^{df}_j \in X^{s+\frac{3}{8}-\epsilon,\frac{1}{2}+}_{|\tau|=|\xi|}[0,T]$ and $\psi_{\pm} \in X^{s,1}_{\pm}[0,T]$. \end{proposition} \begin{corollary}\label{Cor. 2.1} The solution satisfies $\psi \in C^0([0,T],H^s)$, $A^{cf} \in C^0([0,T],H^{s+\frac{1}{8}})$, $|\nabla|^{\epsilon} A^{df} \in C^0([0,T],H^{s+\frac{3}{8}-\epsilon})$ . \end{corollary} \begin{proof}[Proof of Proposition \ref{Prop. 2.1}] We want to apply a Picard iteration. For the Cauchy problem for the Dirac equation $$ (- i\partial_t \pm|\nabla|)\psi_{\pm} = F_{\pm}, \quad \psi_{\pm}(0) = \psi_{\pm 0} $$ we use the well-known estimate (cf. e.g. \cite{GTV}) $$ \|\psi_{\pm}\|_{X^{s,b}_{\pm}[0,T]} \lesssim \|\psi_{\pm 0}\|_{H^s} + T^{1+b'-b} \|F_{\pm}\|_{X^{s,b'}_{\pm}[0,T]} \, , $$ which holds for $03/8$ from the estimate \begin{equation} \label{20} \begin{aligned} \Big|\int v_1 v_2 v_3 dx dt \Big| & \lesssim \|v_1\|_{L^2_x L^{\infty}_t} \|v_2\|_{L^{\infty}_x L^2_t} \|v_3\|_{L^2_x L^2_t} \\ &\lesssim \|v_1\|_{X^{0,\frac{1}{2}+}_{\tau=0}} \|v_2\|_{X^{\frac{1}{2}+, \frac{1}{2}+}_{|\tau|=|\xi|}} \|v_3\|_{X^{0,0}_{|\tau|=|\xi|}} \, , \end{aligned} \end{equation} where we used \eqref{16''}. Case 2: $|\xi_2| \ge |\xi_1|\Rightarrow \langle \xi_3 \rangle^s \lesssim \langle \xi_2 \rangle^s $. In this case the desired estimate follows from \begin{equation}\label{21} \int_* m(\xi_1,\xi_2,\xi_3,\tau_1,\tau_2,\tau_3) \widehat{u}_1(\xi_1,\tau_1) \widehat{u}_2(\xi_2,\tau_2) \widehat{u}_3(\xi_3,\tau_3) d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \end{equation} where $$ m = \frac{1}{ \langle |\tau_2| - |\xi_2|\rangle^{\frac{1}{2}+} \langle \xi_1 \rangle^{\frac{1}{2}+}\langle \tau_1 \rangle^{\frac{1}{2}+}} \, . $$ The following argument is closely related to the proof of a similar estimate in \cite{T1}. By two applications of the averaging principle \cite[Prop. 5.1]{T}, we may replace $m$ by $$ m' = \frac{ \chi_{||\tau_2|-|\xi_2||\sim 1} \chi_{|\tau_1| \sim 1}}{ \langle \xi_1 \rangle^{\frac{1}{2}+}} \, . $$ Let now $\tau_2$ be restricted to the region $\tau_2 =T + O(1)$ for some integer $T$. Then $\tau_3$ is restricted to $\tau_3 = -T + O(1)$, because $\tau_1 + \tau_2 + \tau_3 =0$, and $\xi_2$ is restricted to $|\xi_2| = |T| + O(1)$. The $\tau_3$-regions are essentially disjoint for $T \in {\mathbb Z}$ and similarly the $\tau_2$-regions. Thus by Schur's test \cite[Lemma 3.11]{T}, we only have to show \begin{align*} &\sup_{T \in {\mathbb Z}} \int_* \frac{ \chi_{\tau_3=-T+O(1)} \chi_{\tau_2=T+O(1)} \chi_{|\tau_1|\sim 1} \chi_{|\xi_2| =|T|+O(1)}}{\langle \xi_1 \rangle^{\frac{1}{2}+}} \\ & \times \widehat{u}_1(\xi_1,\tau_1) \widehat{u}_2(\xi_2,\tau_2) \widehat{u}_3(\xi_3,\tau_3) d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} The $\tau$-behaviour of the integral is now trivial, thus we reduce to \begin{equation}\label{50} \sup_{T \in {\mathbb N}} \int_{\sum_{i=1}^3 \xi_i =0} \frac{ \chi_{|\xi_2|=T+O(1)}}{ \langle \xi_1 \rangle^{\frac{1}{2}+}} \widehat{f}_1(\xi_1)\widehat{f}_2(\xi_2)\widehat{f}_3(\xi_3)d\xi \lesssim \prod_{i=1}^3 \|f_i\|_{L^2_x} \, . \end{equation} An elementary calculation shows that \begin{align*} L.H.S. \, of \, \eqref{50} \lesssim \sup_{T \in{\mathbb N}} \| \chi_{|\xi|=T+O(1)} \ast \langle \xi \rangle^{-1-}\|^{1/2}_{L^{\infty}(\mathbb{R}^2)} \prod_{i=1}^3 \|f_i\|_{L^2_x} \lesssim \prod_{i=1}^3 \|f_i\|_{L^2_x}\, , \end{align*} so that the desired estimate follows. \noindent{\bf Proof of \eqref{2.4}:} This reduces to \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^s \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^s \langle |\tau_2| - |\xi_2|\rangle^{\frac{1}{2}+}} \frac{\langle \xi_3 \rangle^{s+\frac{1}{8}} \widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \] Assuming without loss of generality $|\xi_1| \le |\xi_2|$ we have to show \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^s \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{ \langle |\tau_2| - |\xi_2|\rangle^{\frac{1}{2}+}} \frac{\langle \xi_3 \rangle^{1/8} \widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \] Case 1: $|\tau_2| \ll |\xi_2|$. We reduce to \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^s \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{ \langle \xi_2\rangle^{\frac{3}{8}+}} \frac{ \widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \] This follows from \begin{align*} \Big|\int v_1 v_2 v_3 dx dt \Big| & \lesssim \|v_1\|_{L^6_x L^{2+}_t} \|v_2\|_{L^3_x L^2_t} \|v_3\|_{L^2_x L^{\infty -}_t} \\ &\lesssim \|v_1\|_{X^{\frac{1}{6}+,\frac{1}{2}+}_{|\tau|=|\xi|}} \|v_2\|_{X^{\frac{1}{3},0}_{|\tau|=|\xi|}} \|v_3\|_{X^{0,\frac{1}{2}-}_{\tau =0}} \, , \end{align*} where we used \eqref{18} for the first factor and Sobolev for the others. Obviously here is some headroom left. Case 2: $|\tau_2| \gtrsim |\xi_2|$. In this case we use $\tau_1 + \tau_2 + \tau_3 =0$ to estimate $$ 1 \lesssim \frac{\langle \tau_2 \rangle^{\frac{1}{2}-}}{\langle \xi_2 \rangle^{\frac{1}{2}-}} \lesssim \frac{\langle \tau_1 \rangle^{\frac{1}{2}-}}{\langle \xi_2 \rangle^{\frac{1}{2}-}} + \frac{\langle \tau_3 \rangle^{\frac{1}{2}-}}{\langle \xi_2 \rangle^{\frac{1}{2}-}} \, . $$ 2.1: If the second term on the right hand side is dominant we have to show, using also $\langle \xi_3 \rangle^{1/8} \lesssim \langle \xi_2 \rangle^{1/8}$ : \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1 \rangle^s \langle |\tau_1| - |\xi_1|\rangle^{\frac{1}{2}+} } \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2\rangle^{\frac{3}{8}-} \langle |\tau_2|-|\xi_2|\rangle^{\frac{1}{2}+}} \widehat{u}_3(\tau_3,\xi_3) d\xi d\tau \lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \] which follows for $s>\frac{3}{8}$ by Prop. \ref{Prop. 1.1}. 2.2: If the first term on the right hand side is dominant we consider two subcases. 2.2.1: $|\tau_1| \lesssim |\xi_1|$. We reduce to \begin{align*} \int_* \frac{\widehat{u}_1(\tau_1,\xi_1) \langle \xi_1 \rangle^{\frac{1}{2}-s-}}{\langle |\tau_1| - |\xi_1|\rangle^{\frac{1}{2}+} } \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{3}{8}-} \langle |\tau_2|-|\xi_2|\rangle^{\frac{1}{2}+}} \frac{\widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} Using $|\xi_2| \ge |\xi_1|$ and $ s > \frac{3}{8} $ it suffices to show \begin{align*} \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1 \rangle^{\frac{1}{8}+}\langle |\tau_1| - |\xi_1|\rangle^{\frac{1}{2}+} } \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2\rangle^{\frac{1}{8}+} \langle |\tau_2|-|\xi_2|\rangle^{\frac{1}{2}+}} \frac{\widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} This follows from \begin{align*} \Big| \int v_1 v_2 v_3 dx dt \Big| & \lesssim \|v_1\|_{L^4_x L^{2+}_t} \|v_2\|_{L^4_x L^{2+}_t} \|v_3\|_{L^2_x L^{\infty-}_t} \\ &\lesssim \|v_1\|_{X^{\frac{1}{8}+,\frac{3}{8}+}_{|\tau|=|\xi|}} \|v_2\|_{X^{\frac{1}{8}+,\frac{3}{8}+}_{|\tau|=|\xi|} } \|v_3\|_{X^{0,\frac{1}{2}-}_{\tau =0}} \, , \end{align*} where we used \eqref{17}. 2.2.2: $|\tau_1| \gg |\xi_1|\Rightarrow \langle |\tau_1| - |\xi_1| \rangle^{\frac{1}{2}-} \sim \langle \tau_1 \rangle^{\frac{1}{2}-}$. We have to show \begin{align*} \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1 \rangle^s} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2\rangle^{\frac{3}{8}-} \langle |\tau_2|-|\xi_2|\rangle^{\frac{1}{2}+}} \frac{\widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} This follows from \begin{align*} \Big| \int v_1 v_2 v_3 dx dt \Big| & \lesssim \|v_1\|_{L^3_x L^2_t} \|v_2\|_{L^6_x L^{2+}_t} \|v_3\|_{L^2_x L^{\infty-}_t} \\ &\lesssim \|v_1\|_{X^{\frac{1}{3},0}_{|\tau|=|\xi|}} \|v_2\|_{X^{\frac{1}{6}+,\frac{1}{2}+}_{|\tau|=|\xi|} } \|v_3\|_{X^{0,\frac{1}{2}-}_{\tau =0}} \, , \end{align*} where we used Sobolev for the first and last factor and \eqref{18} for the second one. This completes the proof of \eqref{2.4}. \noindent{\bf Proof of \eqref{2.3}:} We distinguish between low and high frequencies of $A_j^{df}$. For high frequencies, i.e. , $\operatorname{supp} (\mathcal{F} A_j^{df}) \subset \{|\xi| \ge 1 \}$, we obtain by \eqref{6} and Prop. \ref{Prop. 1.1} for $s > 3/8$: $$ \| |\nabla|^{\epsilon} A_j^{df} \|_{X^{s+\frac{3}{8}-\epsilon, \frac{1}{2}+}_{|\tau|=|\xi|}} \lesssim \| \langle \psi,\psi \rangle \|_{X^{s-\frac{5}{8},\frac{1}{2}+}_{|\tau|=|\xi|}} \lesssim \|\psi\|^2_{X^{s,\frac{1}{2}++}_{|\tau|=|\xi|}} \, . $$ In the low frequency case $|\xi_3| \le 1$, where $\langle \xi_1 \rangle \sim \langle \xi_2 \rangle$, it suffices to show \begin{align*} &\int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1 \rangle^s \langle |\tau_1|-|\xi_1|\rangle^{\frac{1}{2}++}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2\rangle^s \langle |\tau_2|-|\xi_2|\rangle^{\frac{1}{2}++}} \\ & \times \frac{\widehat{u}_3(\tau_3,\xi_3) \langle |\tau_3|-|\xi_3| \rangle^{\frac{1}{2}+} \langle \xi_3 \rangle^{s+\frac{3}{8}-\epsilon} \chi_{\{|\xi_3| \le 1 \}}}{|\xi_3|^{1-\epsilon}} d\xi d\tau \lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} Assuming without loss of generality $\langle \tau_2 \rangle \le \langle \tau_1 \rangle$, we obtain $ \langle |\tau_3|-|\xi_3| \rangle^{\frac{1}{2}+} \sim \langle \tau_3 \rangle^{\frac{1}{2}+} \lesssim \langle \tau_1 \rangle^{\frac{1}{2}+} + \langle \tau_2 \rangle^{\frac{1}{2}+} \lesssim \langle \tau_1 \rangle^{\frac{1}{2}+} $. If $|\tau_1| \gg |\xi_1|$ or $|\tau_1| \ll |\xi_1|$ , it suffices to show \begin{align*} \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1 \rangle^s} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2\rangle^s \langle |\tau_2|-|\xi_2|\rangle^{\frac{1}{2}++}} \frac{\widehat{u}_3(\tau_3,\xi_3) \chi_{\{|\xi_3| \le 1 \}}}{|\xi_3|^{1-\epsilon}} d\xi d\tau \lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} This follows from $$ \Big|\int v_1 v_2 v_3 dx dt \Big| \lesssim \|v_1\|_{L^2_{xt}} \|v_2\|_{L^{\infty}_t L^2_x} \|v_3\|_{L^2_t L^{\infty}_x} \, , $$ which gives the desired result using $\dot{H}^{1-\epsilon}_x \hookrightarrow L^{\infty}_x$ for low frequencies. If $|\tau_1| \sim |\xi_1|$, we use $\langle \xi_1 \rangle \sim \langle \xi_2 \rangle$ and reduce to \begin{align*} \int_* \widehat{u}_1(\tau_1,\xi_1) \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{2s-\frac{1}{2}-} \langle \langle |\tau_2|-|\xi_2|\rangle^{\frac{1}{2}++}} \frac{\widehat{u}_3(\tau_3,\xi_3) \chi_{\{|\xi_3| \le 1 \}}}{|\xi_3|^{1-\epsilon}} \lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \end{align*} which can be shown as before. We remark that we only used $s>1/4$ in the low frequency case. \noindent{\bf Proof of \eqref{2.2}:} We even prove the estimate with $X^{s,-\frac{1}{2}++}_{|\tau|=|\xi|}$ replaced by $X^{s,0}_{|\tau|=|\xi|}$ on the left hand side. For high frequencies of $A^{df}_j$ we have to show $$ \|A^{df}_j \alpha^j \psi \|_{X^{s,0}_{|\tau|=|\xi|}} \lesssim \|A^{df}\|_{X^{s+\frac{3}{8},\frac{1}{2}+}_{|\tau|=|\xi|}} \|\psi\|_{X^{s,\frac{1}{2}+}_{|\tau|=|\xi|}} \, , $$ which follows by Proposition \ref{Prop. 1.1}. For the low frequency case of $A^{df}_j$ it suffices to show \begin{align*} &\int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1 \rangle^s \langle |\tau_1|-|\xi_1|\rangle^{\frac{1}{2}+}} \widehat{u}_2(\tau_2,\xi_2) \langle \xi_2\rangle^s \frac{\widehat{u}_3(\tau_3,\xi_3) \chi_{\{|\xi_3| \le 1 \}}}{|\xi_3|^{\epsilon}\langle \|\tau_3|-|\xi_3| \rangle^{\frac{1}{2}+} \langle \xi_3 \rangle^{s+\frac{3}{8}-\epsilon}} d\xi d\tau\\ &\lesssim\prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} Using $\langle \xi_1 \rangle \sim \langle \xi_2 \rangle$ and $\langle \xi_3 \rangle \sim 1$ and $\dot{H}^{\epsilon}_x \hookrightarrow L^{\infty}_x$ for low frequencies this easily follows from the estimate $$ \Big|\int v_1 v_2 v_3 dx dt \Big| \lesssim \|v_1\|_{L^{\infty}_t L^2_x} \|v_2\|_{L^2_t L^2_x} \|v_3\|_{L^2_t L^{\infty}_x} \, . $$ This completes the proof of \eqref{2.1}-\eqref{2.4}. The property $\psi_{\pm} \in X^{s,1}_{\pm}[0,T]$ follows immediately from the proof of \eqref{2.1} and \eqref{2.2}. \end{proof} \section{Proof of Theorem \ref{Theorem 1.2}} \begin{proof} Assume $s > 19/40$ , say $s = \frac{19}{40} + \delta$ with $1 \gg \delta > 0$. Let $\psi \in C^0([0,T],H^s)$, $A_j \in C^0([0,T],H^{s+\frac{1}{8}})$. \noindent{\bf Claim 1:} $\psi_{\pm} \in X_{\pm}^{\frac{1}{4}+\alpha,\frac{1}{2}+}[0,T] $, where $\alpha = \frac{1}{40} + \frac{3}{2}\delta-$. By Sobolev's multiplication law we obtain $$ \|A_j \alpha^j \psi_{\pm}\|_{L^2([0,T],H^{2s-\frac{7}{8}})} \lesssim \|A\|_{C^0([0,T],H^{s+\frac{1}{8}})} \|\psi\|_{C^0([0,T],H^s)} T^{1/2} < \infty \, . $$ Thus $\psi_{\pm} \in X^{2s-\frac{7}{8},1}_{\pm}[0,T]$. Interpolation with $\psi_{\pm} \in X_{\pm}^{s,0}[0,T] \subset C^0([0,T],H^s)$ gives $\psi_{\pm} \in X_{\pm}^{\frac{1}{4}+\frac{3}{2}s - \frac{11}{16}-,\frac{1}{2}+}[0,T] = X_{\pm}^{\frac{1}{4}+\alpha,\frac{1}{2}+}[0,T]$. We now iteratively improve the regularity of $\psi_{\pm}$, $A^{cf}$ and $A^{df}$ in order to end up in a class where uniqueness holds by Theorem \ref{Theorem 1.1}. Let us assume that $\psi_{\pm} \in X_{\pm}^{\min(\frac{1}{4}+\alpha_k,s),\frac{1}{2}+}[0,T]$ with $\alpha_k = \frac{1}{40} +\big(\frac{3}{2}\big)^k \delta - $ for some $k \in {\mathbb N}$. This was just shown for $k=1$. If $\frac{1}{4}+\alpha_k \ge s$, we obtain by \eqref{2.3} and \eqref{2.4} $|\nabla|^{\epsilon} A^{df}_j \in X^{s+\frac{3}{8}-\epsilon, \frac{1}{2}+}_{|\tau|=|\xi|}$ and $A^{cf}_j \in X^{s+\frac{1}{8},\frac{1}{2}+}_{\tau=0}[0,T]$, so that uniqueness follows from Theorem \ref{Theorem 1.1}. Otherwise we now prove \noindent{\bf Claim 2:} $A^{cf}_j \in X^{\min(\frac{1}{4}+2\alpha_k-, s+\frac{1}{8}),\frac{1}{2}+}_{\tau=0}[0,T]$. This reduces to $$ \|\langle \psi,\alpha^j \psi \rangle \|_{X^{\frac{1}{4}+2\alpha_k-, -\frac{1}{2}++}_{\tau=0}} \lesssim \|\psi\|_{X^{\frac{1}{4} + \alpha_k,\frac{1}{2}+}_{|\tau|=|\xi|}}^2 \, , $$ which is equivalent to \begin{align*} &\int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^{\frac{1}{4}+\alpha_k} \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{4}+\alpha_k}\langle |\tau_2| - |\xi_2|\rangle^{\frac{1}{2}+}} \frac{\langle \xi_3 \rangle^{\frac{1}{4}+2\alpha_k -} \widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \\ &\lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} Assuming without loss of generality $|\xi_1| \le |\xi_2|$ we reduce to \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^{\frac{1}{4}+\alpha_k} \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle |\tau_2| - |\xi_2|\rangle^{\frac{1}{2}+}} \frac{\langle \xi_3 \rangle^{\alpha_k -} \widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \] Case 1: $|\tau_2| \ll |\xi_2|$. The left hand side is bounded by \begin{align*} & \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^{\frac{1}{4}+\alpha_k} \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2\rangle^{\frac{1}{2}-\alpha_k+}} \frac{ \widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \\ & \lesssim \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^{\frac{3}{4}} \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \widehat{u}_2(\tau_2,\xi_2) \frac{ \widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \end{align*} because by \eqref{16'''} $$ \Big|\int v_1 v_2 v_3 dx dt \Big| \lesssim \|v_1\|_{L^{\infty}_x L^{2+}_t} \|v_2\|_{L^2_{xt}} \|v_3\|_{L^2_x L^{\infty -}_t} \lesssim \|v_1\|_{X^{\frac{1}{2}+,\frac{1}{2}+}_{|\tau|=|\xi|}} \|v_2\|_{X^{0,0}_{|\tau|=|\xi|}} \|v_3\|_{^{0,\frac{1}{2}-}_{\tau =0}} $$ Case 2: $|\tau_2| \gtrsim |\xi_2|$. In this case we obtain $$ 1 \lesssim \frac{\langle \tau_2 \rangle^{\frac{1}{2}-}}{\langle \xi_2 \rangle^{\frac{1}{2}-}} \lesssim \frac{\langle \tau_1 \rangle^{\frac{1}{2}-}}{\langle \xi_2 \rangle^{\frac{1}{2}-}} + \frac{\langle \tau_3 \rangle^{\frac{1}{2}-}}{\langle \xi_2 \rangle^{\frac{1}{2}-}} \, . $$ 2.1: Concerning the second term we use $\langle \xi_3 \rangle \lesssim \langle \xi_2 \rangle$ and reduce to \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1\rangle^{\frac{1}{4}+\alpha_k} \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{2}-\alpha_k+}\langle |\tau_2| - |\xi_2|\rangle^{\frac{1}{2}+}} \widehat{u}_3(\tau_3,\xi_3) d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \] which follows from Proposition \ref{Prop. 1.1}. 2.2: Concerning the first term we consider two subcases. 2.2.1: $|\tau_1| \lesssim |\xi_1|$. This follows from \begin{align*} & \int_* \frac{\widehat{u}_1(\tau_1,\xi_1) \langle \xi_1\rangle^{\frac{1}{4}-\alpha_k-}} {\langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{2} -\alpha_k}\langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \\ &\lesssim \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1\rangle^{\frac{1}{8}+} \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{8}+} \langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \\ &\lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} Here we used $|\xi_1| \le |\xi_2|$ and $\alpha_k < 1/4$, and the last step uses \eqref{17} just as in the proof of \eqref{2.4}. 2.2.2: $|\tau_1| \gg |\xi_1| \Longrightarrow \langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}-} \sim \langle \tau_1 \rangle^{\frac{1}{2}-}$. We reduce to \begin{align*} &\int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1\rangle^{\frac{1}{4}+\alpha_k}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{2}-\alpha_k}\langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_3(\tau_3,\xi_3)}{\langle \tau_3 \rangle^{\frac{1}{2}-}} d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} This is implied by \begin{align*} \Big|\int v_1 v_2 v_3 dx dt \Big| &\lesssim \|v_1\|_{L^{\frac{8}{3}}_x L^2_t} \|v_2\|_{L^8_x L^{2+}_t} \|v_3\|_{L^2_x L^{\infty -}_t} \\ &\lesssim \|v_1\|_{X^{\frac{1}{4},0}_{|\tau|=|\xi|}} \|v_2\|_{X^{\frac{1}{4}+,\frac{1}{2}+}_{|\tau|=|\xi|}} \|v_3\|_{X^{0,\frac{1}{2}-}_{\tau =0}} \, , \end{align*} where we used Sobolev, \eqref{19} and $\alpha_k < 1/4$. \noindent{\bf Claim 3:} $|\nabla|^{\epsilon} A^{df}_j \in X^{\frac{1}{2}+2\alpha_k - \epsilon -,\frac{1}{2}+}_{|\tau|=|\xi|} $. For high frequencies we obtain $$ \||\nabla|^{\epsilon} A^{df}_j\|_{X^{\frac{1}{2} +2\alpha_k-\epsilon-, \frac{1}{2}+}} \lesssim \| \langle \psi,\psi \rangle\|_{X^{-\frac{1}{2} +2\alpha_k-,\frac{1}{2}+}} \lesssim \|\psi\|^2_{X^{\frac{1}{4} +\alpha_k,\frac{1}{2}++}} $$ by use of Proposition \ref{Prop. 1.1}. The low frequency case can be handled as in the proof of \eqref{2.3}. If after such an iteration step we obtained an $\alpha_k$ such that $\alpha_k > \frac{1}{8}$, we obtain by \eqref{2.1} and \eqref{2.2} combined with claim 2 and claim 3 the regularity $\psi_{\pm} \in X^{\frac{1}{4}+\alpha_k,\frac{1}{2}+}_{\pm}[0,T] \subset X^{\frac{3}{8}+,\frac{1}{2}+}_{\pm}[0,T]$, $|\nabla|^{\epsilon} A^{df}_j \in X^{\frac{1}{2}+2\alpha_k - \epsilon -, \frac{1}{2}+}_{|\tau|=|\xi|} \subset X^{\frac{3}{4}-\epsilon, \frac{1}{2}+}_{|\tau|=|\xi|}[0,T] $ and $A^{cf}_j \in X^{\frac{1}{4}+2\alpha_k-,\frac{1}{2}+}_{\tau=0}[0,T] \subset X^{\frac{1}{2}+,\frac{1}{2}+}_{\tau =0}[0,T]$, where uniqueness holds by Theorem \ref{Theorem 1.1} and we are done. If however $\alpha_k \le \frac{1}{8}$ we need a further iteration step. \noindent{\bf Claim 4:} The following estimate holds: $$ \|A^{cf}_j \alpha^j \psi_{\pm}\|_{L^2_t(H_x^{3\alpha_k --})} \lesssim \|A^{cf}\|_{X^{\frac{1}{4}+2\alpha_k-,\frac{1}{2}+}_{\tau=0}} \|\psi_{\pm}\|_{X^{\frac{1}{4}+\alpha_k,\frac{1}{2}+}_{\pm}} \, . $$ This reduces to \begin{align*} & \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{ \langle \xi_1\rangle^{\frac{1}{4}+2\alpha_k-}\langle \tau_1 \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{4}+\alpha_k}\langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \widehat{u}_3(\tau_3,\xi_3) \langle \xi_3 \rangle^{3\alpha_k--} d\xi d\tau \\ & \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, . \end{align*} Case 1: $|\xi_1| \ge |\xi_2|\Rightarrow\langle \xi_3 \rangle \lesssim \langle \xi_1 \rangle$. Using $\alpha_k \le 1/4$ it suffices to show \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \tau_1 \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{2}+} \langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \widehat{u}_3(\tau_3,\xi_3) d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \] which holds by \eqref{20}. Case 2: $|\xi_2| \ge |\xi_1|\Rightarrow\langle \xi_3 \rangle \lesssim \langle \xi_2 \rangle$. Here we use $\alpha_k \le \frac{1}{8}$. We only have to show \[ \int_* \frac{\widehat{u}_1(\tau_1,\xi_1)}{\langle \xi_1 \rangle^{\frac{1}{2}+} \langle \tau_1 \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \widehat{u}_3(\tau_3,\xi_3) d\xi d\tau \lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \] which follows from \eqref{21}. \noindent{\bf Claim 5:} The following estimate holds: $$ \|A^{df}_j \alpha^j \psi_{\pm}\|_{L^2_t(H_x^{3\alpha_k --})} \lesssim \||\nabla|^{\epsilon} A^{df}\|_{X^{\frac{1}{2}+2\alpha_k-\epsilon-, \frac{1}{2}+}_{|\tau|=|\xi|}} \|\psi_{\pm}\|_{X^{\frac{1}{4}+\alpha_k, \frac{1}{2}+}_{\pm}} \, . $$ In the case of high frequencies of $A^{df}_j$ this follows from Proposition \ref{Prop. 1.1}, where we have to use our assumption $\alpha_k \le \frac{1}{8}$. In the case of low frequencies we can reduce to \begin{align*} &\int_* \frac{\widehat{u}_1(\tau_1,\xi_1) \chi_{\{|\xi_1| \le 1 \}}} {|\xi_1|^{\epsilon}\langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{4}+\alpha_k}\langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \widehat{u}_3(\tau_3,\xi_3) \langle \xi_3 \rangle^{3\alpha_k --} d\xi d\tau \\ &\lesssim \int_* \frac{\widehat{u}_1(\tau_1,\xi_1) \chi_{\{|\xi_1| \le 1 \}}} {|\xi_1|^{\epsilon}\langle |\tau_1|-|\xi_1| \rangle^{\frac{1}{2}+}} \frac{\widehat{u}_2(\tau_2,\xi_2)}{\langle \xi_2 \rangle^{\frac{1}{4}-2\alpha_k}\langle |\tau_2|-|\xi_2| \rangle^{\frac{1}{2}+}} \widehat{u}_3(\tau_3,\xi_3) d\xi d\tau \\ &\lesssim \prod_{i=1}^3 \|u_i\|_{L^2_{xt}} \, , \end{align*} which easily follows from the estimate $$ \big| \int v_1 v_2 v_3 dx dt \big| \lesssim \|v_1\|_{L^{\infty}_{xt}} \|v_2\|_{L^2_{xt}} \|v_3\|_{L^2_{xt}} $$ for low frequencies of $v_1$, where we used again $\alpha_k \le 1/8$. We recall that $\alpha_k = \frac{1}{40}+\big(\frac{3}{2}\big)^k \delta \to \infty $ $(k \to \infty) $ and $s=\frac{19}{40}+\delta$ with $1 \gg \delta >0$. Thus for some $k \in {\mathbb N}$ we have $\alpha_k \le \frac{1}{8}$ and $\alpha_{k+1} > \frac{1}{8}$. Claim 4 and claim 5 imply that $\psi_{\pm} \in X_{\pm}^{\min(3\alpha_k-,s),1}[0,T]$. Interpolation with $\psi_{\pm} \in X_{\pm}^{s,0}[0,T] \supset C^0([0,T],H^s)$ gives $\psi_{\pm} \in X_{\pm}^{\min(\frac{3}{2} \alpha_k+\frac{s}{2}-,s), \frac{1}{2}+}[0,T]$. We notice that $\frac{3}{2}\alpha_k + \frac{s}{2} = \frac{1}{4} + \big( \frac{1}{40} + \big(\frac{3}{2}\big)^{k+1} \delta) + \frac{\delta}{2} > \frac{1}{4}+\alpha_{k+1}$. Therefore $\psi_{\pm} \in X_{\pm}^{\min(\frac{1}{4}+\alpha_{k+1},s),\frac{1}{2}+}[0,T] \subset X_{\pm}^{\frac{3}{8}+,\frac{1}{2}+}[0,T] $, and by \eqref{2.3} and \eqref{2.4} we obtain $A^{cf}_j \in X^{\frac{1}{2}+, \frac{1}{2}+}_{\tau = 0}[0,T]$ and $|\nabla|^{\epsilon} A^{df} \in X_{\pm}^{\frac{3}{4}-\epsilon+,\frac{1}{2}+}[0,T]$. In these spaces however uniqueness holds by Theorem \ref{Theorem 1.1}. \end{proof} \section{Proof of Theorem \ref{Theorem 1.1'} and Theorem \ref{Theorem 1.2'}} \begin{proof}[Proof of Theorem \ref{Theorem 1.1'}] By standard arguments we only have to show $$ \| N(\psi_1,\psi_2,\psi_3) \|_{X^{s,-\frac{1}{2}++}_{\pm_4}} \lesssim \prod_{i=1}^3 \|\psi_i\|_{X^{s,\frac{1}{2}+}_{\pm_ i}} \, , $$ where $\pm_i$ $(i=1,2,3,4)$ denote independent signs. By duality this is reduced to the estimates $$ J:= \int \langle N(\psi_1,\psi_2,\psi_3), \psi_4 \rangle dt\, dx \lesssim \prod_{i=1}^3 \|\psi_i\|_{X^{s,\frac{1}{2}+}_{\pm_i}} \|\psi_4\|_{X^{-s,\frac{1}{2}--}_{\pm_4}} \, . $$ By Fourier-Plancherel we obtain $$ J = \int_* q(\xi_1,\dots ,\xi_4) \prod_{j=1}^4 \widehat{\psi}_j(\xi_j,\tau_j) d\xi_1\, d\tau_1 \dots d\xi_4\,d\tau_4 \, , $$ where * denotes integration over $\xi_1-\xi_2=\xi_4-\xi_3=:\xi_0$ and $\tau_1-\tau_2=\tau_4-\tau_3$ and \begin{align*} q = \frac{1}{|\xi_0|^2} &[( \xi_{0_1}(\langle \widehat{\psi}_1, \alpha_2 \widehat{\psi}_2 \rangle \langle \widehat{\psi}_3,\widehat{\psi}_4\rangle - \langle \widehat{\psi}_1,\widehat{\psi}_2 \rangle \langle \alpha_2 \widehat{\psi}_3,\widehat{\psi}_4\rangle) \\ & - \xi_{0_2}(\langle \widehat{\psi}_1,\alpha_1 \widehat{\psi}_2 \rangle \langle \widehat{\psi}_3,\widehat{\psi}_4\rangle - \langle \widehat{\psi}_1,\widehat{\psi}_2 \rangle \langle \alpha_1 \widehat{\psi}_3,\widehat{\psi}_4\rangle) ] \, . \end{align*} The specific structure of this term, namely the form of the matrices $\alpha_j$ plays no role in the following, thus the null structure is completely ignored. We first consider the case $|\xi_0| \le 1$. In this case we estimate $J$ as follows: \begin{align*} &\| \langle \nabla \rangle^{-s-1} |\nabla|^{-\frac{1}{2}} \langle \psi_1,\alpha_i\psi_2 \rangle \|_{L^2_{xt}} \lesssim \| \langle \psi_1,\alpha_i\psi_2 \rangle \|_{L^2_{x} H^{-s-1,\frac{4}{3}}_x} \lesssim \|\psi_1\|_{L^4_t H^s_x} \|\psi_2\|_{L^4_t H^{-s}_x} \, . \end{align*} In the last step we used \begin{align*} \| f g \|_{H^s_x} \lesssim \|f\|_{H^s_x} \|g\|_{L^{\infty}_x} + \|f\|_{L^2_x} \|g\|_{H^{s,\infty}_x} \lesssim \|f\|_{H^s_x} \|g\|_{H^{s+1,4}_x} \, \end{align*} which holds by the Leibniz rule for fractional derivatives and Sobolev's embedding theorem, and which is by duality equivalent to the required estimate $$ \|fg\|_{H^{-s-1,\frac{4}{3}}_x} \lesssim \|f\|_{H^s_x} \|g\|_{H^{-s}_x} \, . $$ The same estimate holds for $\alpha_i = I$. Similarly we obtain \begin{align*} &\| \langle \nabla \rangle^{-s-1} |\nabla|^{-\frac{1}{2}} \langle \alpha_i \psi_3,\psi_4 \rangle \|_{L^2_{xt}} \lesssim \| \psi_3\|_{L^4_t H^s_x} \| \psi_4\|_{L^4_t H^{-s}_x} \end{align*} for arbitrary matrices $\alpha_i$, so that we obtain \begin{align*} J & \lesssim \|\psi_1\|_{X^{s,\frac{1}{4}}_{\pm_1}} \|\psi_2\|_{X^{-s, \frac{1}{4}}_{\pm_2}} \|\psi_3\|_{X^{s,\frac{1}{4}}_{\pm_3}} \|\psi_4\|_{X^{-s,\frac{1}{4}}_{\pm_4}} \, , \end{align*} which is more than enough. From now on we assume $|\xi_0| \ge 1$. We obtain \begin{align*} |J| &\lesssim \sum_{j=1}^2 \big( \|\langle \psi_1,\alpha_j\psi_2 \rangle \|_{X_{|\tau|=|\xi|}^{s-\frac{1}{2},\frac{1}{4}}} \|\langle \psi_3,\psi_4\rangle\|_{X_{|\tau|=|\xi|}^{-s-\frac{1}{2},-\frac{1}{4}}} \\ & \quad+ \|\langle \psi_1,\psi_2 \rangle \|_{X_{|\tau|=|\xi|}^{s-\frac{1}{2}, \frac{1}{4}}} \|\langle \alpha_j \psi_3, \psi_4\rangle\|_{X_{|\tau|=|\xi|}^{-s-\frac{1}{2},-\frac{1}{4}}} \big)\, . \end{align*} By Proposition \ref{Prop. 1.1} with $s_0=\frac{1}{2}-s$, $b_0=-1/4$, $s_1=s_2 =s$, $b_1=b_2=\frac{1}{2}+\epsilon$ for the first factors and $s_0 = s+\frac{1}{2}$, $b_0 = 1/4$, $s_1=s$, $s_2=-s$, $b_1=\frac{1}{2}+\epsilon$, $b_2=\frac{1}{2}-2\epsilon$ for the second factors we obtain $$ |J| \lesssim \prod_{j=1}^3 \|\psi_j\|_{X_{|\tau| =|\xi|}^{s,\frac{1}{2}+\epsilon}} \|\psi_4\|_{X_{|\tau|=|\xi|}^{-s,\frac{1}{2} -2\epsilon}} \, . $$ Using the embedding $X^{s,b}_{\pm} \subset X_{|\tau|=|\xi|}^{s,b}$ for $s \in \mathbb{R}$ and $b \ge 0$ we obtain the desired estimate. \end{proof} \noindent{\bf Remark:} The potentials are completely determined by $\psi$ and \eqref{1.10}. We have $A_{\mu} \sim |\nabla|^{-1} \langle \psi, \psi \rangle$, so that for $s\le \frac{1}{2}$: $$ \|A_{\mu}\|_{\dot{H}^{2s}} \lesssim \|\langle \psi,\psi\rangle \|_{\dot{H}^{2s-1}} \lesssim \|\langle \psi,\psi\rangle \|_{L^{\frac{1}{1-s}}} \lesssim \|\psi\|^2_{L^{\frac{2}{1-s}}} \lesssim \|\psi\|^2_{H^s} < \infty $$ and for $\frac{1}{2} < s < 1$: $$ \|A_{\mu}\|_{\dot{H}^{2s}} \lesssim \|\langle \psi,\psi\rangle \|_{\dot{H}^{2s-1}} \lesssim \| \psi \|_{\dot{H}^{2s-1,\frac{2}{s}}} \|\psi\|_{L^{\frac{2}{1-s}}} \lesssim \|\psi\|_{H^s}^2 < \infty $$ as well as $$ \|A_{\mu}\|_{\dot{H}^{\epsilon}} \lesssim \|\langle \psi,\psi\rangle \|_{\dot{H}^{\epsilon -1}} \lesssim \|\psi\|^2_{L^{\frac{4}{2-\epsilon}}} \lesssim \|\psi\|^2_{H^s} < \infty \, , $$ thus we obtain for $0<\epsilon\ll 1$ and $s<1$: $$ A_{\mu} \in C^0([0,T],\dot{H}^{2s} \cap \dot{H}^{\epsilon}) \,. $$ \begin{proof}[Proof of Theorem \ref{Theorem 1.2'}] We first show $\psi_{\pm} \in X^{0,1}_{\pm}[0,T]$. We have to prove $$ \|N(\psi_1,\psi_2,\psi_3)\|_{L^2_t([0,T],L^2_x)} \lesssim \prod_{j=1}^3 \|\psi_j\|_{L^{\infty}_t([0,T], H^{\frac{1}{3}}_x)} \, , $$ where the implicit constant may depend on $T$. This follows from the estimate \begin{align*} \| |\nabla|^{-1} \langle \psi_j,\alpha_i \psi_k \rangle \psi_3\|_{L^2_x} & \lesssim \| |\nabla|^{-1} \langle \psi_j,\alpha_i\psi_k \rangle \|_{L^6_x} \|\psi_3\|_{L^3_x} \\ &\lesssim \| \langle \psi_j,\alpha_i\psi_k \rangle \|_{L^{\frac{3}{2}}_x} \|\psi_3\|_{L^3_x} \\ & \lesssim \|\psi_j\|_{L^3_x} \|\psi_k\|_{L^3_x} \|\psi_3\|_{L^3_x}\\ & \lesssim \|\psi_j\|_{H^{\frac{1}{3}}_x} \|\psi_k\|_{H^{\frac{1}{3}}_x} \|\psi_3\|_{H^{\frac{1}{3}}_x} \, , \end{align*} and a similar estimate for the term $\| |\nabla|^{-1} \langle \psi_j,\psi_k \rangle \alpha_i \psi_3\|_{L^2_x}$. Assume now $\psi \in C^0([0,T],H^{\frac{1}{3}+\epsilon})$, $\epsilon > 0$. Then we have shown that $\psi_{\pm} \in X_{\pm}^{\frac{1}{3}+\epsilon,0}[0,T] \cap X^{0,1}_{\pm}[0,T]$. By interpolation we get $\psi_{\pm} \in X^{\frac{1}{4}+\frac{\epsilon}{4},\frac{1}{4}+\epsilon}_{\pm} [0,T] $ for $ \epsilon \ll 1$. Assume now that $\psi,\psi' \in C^0([0,T],H^{\frac{1}{3}+\epsilon})$ are two solutions of \eqref{1.11},\eqref{1.12}, Then we have \begin{equation} \label{**} \begin{aligned} &\sum_{\pm} \|\psi_{\pm}-\psi_{\pm}'\|_{X^{0,\frac{1}{2}+}_{\pm}[0,T]} \\ &\lesssim T^{0+} \sum_{\pm} \|N(\psi,\psi,\psi)-N(\psi',\psi',\psi') \|_{X_{\pm}^{0,-\frac{1}{2}++}[0,T]}\\ & \lesssim T^{0+} \hspace{-1em} \sum_{\pm,\pm_1,\pm_2,\pm_3} \big(\|N(\psi_{\pm_1}-\psi_{\pm_1}',\psi_{\pm_2},\psi_{\pm_3}) \|_{X_{\pm}^{0,-\frac{1}{2}++}[0,T]}\\ & \quad+ \|N(\psi_{\pm_1}',\psi_{\pm_2}-\psi_{\pm_2}',\psi_{\pm_3}) \|_{X_{\pm}^{0,-\frac{1}{2}++}[0,T]}\\ &\quad +\|N(\psi_{\pm_1}',\psi_{\pm_2}',\psi_{\pm_3} -\psi_{\pm_3}')\|_{X_{\pm}^{0,-\frac{1}{2}++}[0,T]}\big) \end{aligned} \end{equation} Here $\pm$, $\pm_j$ $(j=1,2,3)$ denote independent signs. We want to show that for the first term the following estimate holds: \begin{equation} \label{*} \begin{aligned} J:= &\int \langle N(\psi_{\pm_1}-\psi_{\pm_1}',\psi_{\pm_2},\psi_{\pm_3}), \psi_{4} \rangle dx\,dt \\ \lesssim& \|\psi_{\pm_1}-\psi_{\pm_1}'\|_{X^{0,\frac{1}{2}+}_{\pm_1}} \|\psi_{\pm_2}\|_{X^{\frac{1}{4}+\frac{\epsilon}{4},\frac{1}{4}+\epsilon}_{\pm_2}} \|\psi_{\pm_3}\|_{X^{\frac{1}{4}+\frac{\epsilon}{4},\frac{1}{4}+\epsilon}_{\pm_3}} \|\psi_{4}\|_{X^{0,\frac{1}{2}--}_{\pm_4}} \, . \end{aligned} \end{equation} We consider the case $|\xi_0|\le 1$ first. Similarly as in the proof of Theorem \ref{Theorem 1.1'} we obtain $$ |J| \lesssim \|\psi_{\pm_1} - \psi_{\pm_1}'\|_{X^{-\frac{1}{4} -\frac{\epsilon}{4},\frac{1}{4}}_{\pm_1}} \|\psi_{\pm_2}\|_{X^{\frac{1}{4} +\frac{\epsilon}{4},\frac{1}{4}}_{\pm_2}} \|\psi_{\pm_3}\|_{X^{\frac{1}{4}+\frac{\epsilon}{4},\frac{1}{4}}_{\pm_3}} \|\psi_{4}\|_{X^{-\frac{1}{4}-\frac{\epsilon}{4},\frac{1}{4}}_{\pm_4}}\, , $$ which is more than sufficient. For $|\xi_0| \ge 1$ we obtain \begin{align*} |J| &\lesssim \sum_{j=1}^2 \big( \| \langle (\psi_{\pm_1} - \psi_{\pm_1}'), \alpha_j \psi_{\pm_2} \rangle \|_{X_{|\tau|=|\xi|}^{-\frac{1}{2},0}} \|\langle \psi_{\pm_3},\psi_4 \rangle\|_{X_{|\tau|=|\xi|}^{-\frac{1}{2},0}}\\ &\quad + \| \langle \psi_{\pm_1} - \psi_{\pm_1}',\psi_{\pm_2} \rangle \|_{X_{|\tau|=|\xi|}^{-\frac{1}{2},0}} \|\langle \alpha_j \psi_{\pm_3},\psi_4 \rangle\|_{X_{|\tau|=|\xi|}^{-\frac{1}{2},0}}\big)\\ & \lesssim \|\psi_{\pm_1} - \psi_{\pm_1}'\|_{X_{|\tau|=|\xi|}^{0,\frac{1}{2}+}} \|\psi_{\pm_2}\|_{X_{|\tau|=|\xi|}^{\frac{1}{4}+\frac{\epsilon}{4}, \frac{1}{4}+\epsilon}} \|\psi_{\pm_3}\|_{X_{|\tau|=|\xi|}^{\frac{1}{4}+\frac{\epsilon}{4}, \frac{1}{4}+\epsilon}} \|\psi_4\|_{X_{|\tau|=|\xi|}^{0,\frac{1}{2}--}} \, , \end{align*} where we used Proposition \ref{Prop. 1.1} for the first factor with the choice $s_0 = \frac{1}{2}$, $b_0=0$, $s_1=0$, $b_1=\frac{1}{2}+$, $s_2=\frac{1}{4}+\frac{\epsilon}{4}$, $b_2 = \frac{1}{4}+\epsilon$ and for the second factor with $s_0 = \frac{1}{2}$, $b_0=0$, $s_1= \frac{1}{4}+\frac{\epsilon}{4}$, $b_1 = \frac{1}{4}+\epsilon$, $s_2=0$, $b_2= \frac{1}{2}--$. The embedding $X^{s,b}_{\pm} \subset X_{|\tau|=|\xi|}^{s,b}$ for $b \ge 0$ gives \eqref{*}. The other terms in \eqref{**} are treated similarly. We obtain \begin{align*} &\sum_{\pm} \|\psi_{\pm} - \psi_{\pm}'\|_{X^{0,\frac{1}{2}+}_{\pm}[0,T]} \\ & \lesssim T^{0+} \sum_{j=1}^2 \big( \|\psi_{\pm_j}\|^2_{X^{\frac{1}{4} +\frac{\epsilon}{4}, \frac{1}{4}+\epsilon}_{\pm_j}[0,T]} +\|\psi_{\pm_j}'\|^2_{X^{\frac{1}{4}+\frac{\epsilon}{4}, \frac{1}{4}+\epsilon}_{\pm_j}[0,T]} \big) \sum_{\pm} \|\psi_{\pm}-\psi_{\pm}'\|_{X^{0,\frac{1}{2}+}_{\pm}[0,T]} \, . \end{align*} We recall that $\psi_{\pm} \, , \, \psi'_{\pm} \in X^{\frac{1}{4} +\frac{\epsilon}{4},\frac{1}{4}+\epsilon}_{\pm} [0,T] $, so that for sufficiently small $T$ this implies $\|\psi_{\pm} - \psi_{\pm}'\|_{X^{0,\frac{1}{2}+}_{\pm}[0,T]}=0$, thus local uniqueness. By iteration $T$ can be chosen arbitrarily. \end{proof} \begin{thebibliography}{00} \bibitem{AFS1} P. d'Ancona, D. Foschi, S. Selberg; \emph{Null structure and almost optimal regularity for the Dirac-Klein-Gordon system}. J. European Math. Soc. 9(4) (2007), 877-899. \bibitem{AFS} P. d'Ancona, D. Foschi, S. Selberg; \emph{Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions}. Contemporary Math. 526 (2010), 125-150. \bibitem{BL} J. Bergh J. L\"ofstr\"om; \emph{Interpolation spaces}. Springer Berlin 1976. \bibitem{BCM} N. Bourneveas, T. Candy, S. Machihara; \emph{A note on the Chern-Simons-Dirac equations in the Coulomb gauge}. Discr. Cont. Dyn. Syst. 34(7) (2014), 2693-2701. \bibitem{CKP} Y. M. Cho, J. W. Kim, D.H. Park; \emph{Fermionic vortex solutions in Chern-Simons electrodynamics}. Phys. Rev. D 45 (1992), 3802-3806. \bibitem{GTV} J. Ginibre, Y. Tsutsumi, G. Velo; \emph{On the Cauchy problem for the Zakharov system}. J. Funct. Anal. 151 (1997), 384-436. \bibitem{H} H. Huh; \emph{Cauchy problem for the Fermion field equation coupled with the Chern-Simons gauge}. Lett. Math. Phys. 79 (2007), 75-94. \bibitem{HO} H. Huh, S.-J. Oh; \emph{Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge}. Comm. PDE 41(3) (2016), 375-397. \bibitem{KMBT} S. Klainerman, M. Machedon (Appendices by J. Bougain and D. Tataru); \emph{Remark on Strichartz-type inequalities}. Int. Math. Res. Notices 1996, no.5, 201-220. \bibitem{KT} S. Klainerman, D. Tataru; \emph{On the optimal local regularity for Yang-Mills equations in $\mathbb{R}^{4+1}$}. J. Amer. Math. Soc. 12(1) (1999), 93-116. \bibitem{LB} S. Li, R. K. Bhaduri; \emph{Planar solitons of the gauged Dirac equation}. Phys. Rev. D 43 (1991), 3573-3574. \bibitem{O} M. Okamoto; \emph{Well-posedness of the Cauchy problem for the Chern-Simons-Dirac system in two dimensions}. J. Hyp. Diff. Equ. 10(4) (2013), 735-771. \bibitem{T} T. Tao; \emph{Multilinear weighted convolution of $L^2$-functions and applications to non-linear dispersive equations}. Amer. J. Math. 123 (2001), 838-908. \bibitem{T1} T. Tao; \emph{Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm}. J. Diff. Equ. 189 (2003), 366-382. \bibitem{Z1} Y. Zhou; \emph{Local existence with minimal regularity for nonlinear wave equations}. Amer. J. Math. 119(3) (1997), 671-703. \bibitem{Z} Y. Zhou; \emph{Uniqueness of generalized solutions to nonlinear wave equations}. Amer. J. Math. 122(5) (2000), 939-965. \end{thebibliography} \end{document}