\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 114, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/114\hfil Chern-Simons-Higgs equations] {Low regularity solutions of the Chern-Simons-Higgs equations in the Lorentz gauge} \author[N. Bournaveas \hfil EJDE-2009/114\hfilneg] {Nikolaos Bournaveas} \address{Nikolaos Bournaveas \newline University of Edinburgh, School of Mathematics, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK} \email{n.bournaveas@ed.ac.uk} \thanks{Submitted February 22, 2009. Published September 12, 2009.} \subjclass[2000]{35L15, 35L70, 35Q40} \keywords{Chern-Simons-Higgs equations; Lorentz gauge; null-form estimates; \hfill\break\indent low regularity solutions} \begin{abstract} We prove local well-posedness for the $2+1$-dimensional Chern-Simons-Higgs equations in the Lorentz gauge with initial data of low regularity. Our result improves earlier results by Huh \cite{Huh1, Huh2}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{claim}[theorem]{Claim} \newcommand{\abs}[1]{|#1|} \newcommand{\norm}[2]{\|#1\|_{#2}} \section{Introduction} The Chern-Simon-Higgs model was proposed by Jackiw and Weinberg \cite{JW} and Hong, Pac and Kim \cite{HKP} in the context of their studies of vortex solutions in the abelian Chern-Simons theory. Local well-posedness of low regularity solutions was recently studied in Huh \cite{Huh1, Huh2} using a null-form estimate for solutions of the linear wave equation due to Foschi and Klainerman \cite{FK} as well as Strichartz estimates. Our aim in this paper is to improve the results of \cite{Huh1, Huh2} in the Lorentz gauge. For this purpose we use estimates in the restriction spaces $X^{s,b}$ introduced by Bourgain, Klainerman and Machedon. A key ingredient in our proof is a modified version of a null-form estimate of Zhou \cite{zhou} and product rules in $X^{s,b}$ spaces due to D'Ancona, Foschi and Selberg \cite{DFS1, DFS2} and Klainerman and Selberg \cite{KS}. The Higgs field has fractional dimension (see below for details), a common feature of systems involving the Dirac equation, see for example Bournaveas \cite{B1, B2}, D'Ancona, Foschi and Selberg \cite{DFS1, DFS2}, Machihara \cite{M1, M2}, Machihara, Nakamura, Nakanishi and Ozawa \cite{MNNO}, Selberg and Tesfahun \cite{ST}, Tesfahun \cite{T}. The Chern-Simon-Higgs equations are the Euler-Lagrange equations corresponding to the Lagrangian density $\mathcal{L}=\frac{\kappa}{4} \epsilon^{\mu\nu\rho} A_{\mu} F_{\nu\rho} + D_{\mu}\phi\, \overline{D^\mu \phi} - V\big(|\phi|^2\big).$ Here $A_{\mu}$ is the gauge field, $F_{\mu\nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ is the curvature, $D_{\mu}=\partial_{\mu}-iA_{\mu}$ is the covariant derivative, $\phi$ is the Higgs field, $V$ is a given positive function and $\kappa$ is a positive coupling constant. Greek indices run through $\{0,1,2\}$, Latin indices run through $\{1,2\}$ and repeated indices are summed. The Minkowski metric is defined by $(g^{\mu\nu})=diag(1,-1,-1)$. We define $\epsilon^{\mu\nu\rho} =0$ if two of the indices coincide and $\epsilon^{\mu\nu\rho}=\pm 1$ according to whether $(\mu,\nu,\rho)$ is an even or odd permutation of $(0,1,2)$. We define Klainerman's null forms by \begin{subequations}\label{NF} \begin{gather} Q_{\mu\nu}(u,v)=\partial_\mu u \partial_\nu v - \partial_\nu u \partial_\mu v, \\ Q_{0}(u,v)= g^{\mu\nu}\partial_\mu u \partial_\nu v . \end{gather} \end{subequations} Let $I^{\mu}=2 Im \big(\overline{\phi} D^{\mu}\phi \big)$. Then the Euler-Lagrange equations are (we set $\kappa=2$ for simplicity) \begin{subequations}\label{EL} \begin{gather} F_{\mu\nu}=\frac{1}{2} \epsilon_{\mu\nu\alpha}I^\alpha ,\\ D_\mu D^\mu \phi = -\phi V'\big(|\phi|^2\big). \end{gather} \end{subequations} The system has the positive conserved energy given by $\mathcal{E}=\int_{\mathbb{R}^2} \sum_{\mu=0}^{2} \abs{D_{\mu}\phi}^2 + V(|\phi|^2) \,dx.$ We are interested in the so-called non-topological' case in which $|\phi|\to 0$ as $|x|\to +\infty$. For the sake of simplicity we follow \cite{Huh1, Huh2} and set $V=0$. It will be clear from our proof that for various classes of $V$'s the term $\phi V'(|\phi|^2)$ can easily be handled. Under the Lorentz gauge condition $\partial^{\mu} A_{\mu}=0$ the Euler-Lagrange equations \eqref{EL} become \begin{subequations}\label{L1} \begin{gather} \partial_0 A_j = \partial_j A_0 + \tfrac12 \epsilon_{ij}I_{i} , \\ \partial_1 A_2 = \partial_2 A_1 + \tfrac12 I_0 , \\ \partial_0 A_0 = \partial_1 A_1 + \partial_2 A_2 , \\ D_\mu \, D^\mu\, \phi = 0 . \end{gather} \end{subequations} Alternatively, they can be written as a system of two nonlinear wave equations: \begin{subequations}\label{L2} \begin{gather} \Box A^\alpha = \frac{1}{2}\epsilon^{\alpha\beta\gamma} \mathop{\rm Im}(\overline{D_\gamma \phi} D_\beta \phi - \overline{D_\beta \phi} D_\gamma \phi ) + \frac{1}{2}\epsilon^{\alpha\beta\gamma} (\partial_\beta A_\gamma - \partial_\gamma A_\beta )|\phi|^2 , \label{L2a}\\ \Box \phi = 2iA^\alpha\partial_\alpha \phi + A^\alpha A_\alpha \phi .\label{L2b} \end{gather} \end{subequations} We prescribe initial data in the classical Sobolev spaces $A^{\mu}(0,x)=a^{\mu}_{0}(x) \in H^{a}$, $\partial_{t} A^{\mu}(0,x)=a^{\mu}_{1}(x)\in H^{a-1}$, $\phi(0,x)=\phi_{0}(x) \in H^{b}$, $\partial_{t} \phi(0,x)=\phi_{1}(x)\in H^{b-1}$. Dimensional analysis shows that the critical values of $a$ and $b$ are $a_{cr}=0$ and $b_{cr}=\frac12$. It is well known that in low space dimensions the Cauchy problem may not be locally well posed for $a$ and $b$ close to the critical values due to lack of decay at infinity. Observe also that $\phi$ has fractional dimension. From the point of view of scaling it is natural to take $b=a+\frac12$. With this choice it was shown in Huh \cite{Huh1} that the Cauchy problem is locally well posed for $a=\frac34+\epsilon$ and $b=\frac54+\epsilon$. This was improved in Huh \cite{Huh2} to $$\label{H1} a=\frac34+\epsilon\,,\quad b=\frac98+\epsilon$$ (slightly violating $b=a+\frac12$). The proof relies on the null structure of the right hand side of \eqref{L2a}. Indeed, $\overline{D_\gamma \phi} D_\beta \phi - \overline{D_\beta \phi} D_\gamma \phi = Q_{\gamma\beta}(\overline{\phi},\phi) + i \left( A_{\gamma}\partial_\beta(|\phi|^2) - A_{\beta}\partial_{\gamma}(|\phi|^2)\right).$ On the other hand, since in \eqref{L1} the $A_\mu$ satisfy first order equations and $\phi$ satisfies a second order equation it is natural to investigate the case $b=a+1$. It turns out that this choice allows us to improve on $a$ at the expense of $b$. It is shown in Huh \cite{Huh2} that we have local well posedness for $$\label{H2} a=\frac12 \,,\quad b=\frac32 .$$ To prove this result Huh uncovered the null structure in the right hand side of equation \eqref{L2b}. Indeed, if we introduce $B_{\mu}$ by $\partial_{\mu} B^{\mu}=0$ and $\partial_{\mu}B_\nu-\partial_{\nu}B_\mu=\epsilon_{\mu\nu\lambda} A^{\lambda}$, then the equations take the form: \begin{subequations}\label{L3} \begin{gather} \Box B^\gamma =-\mathop{\rm Im} \left(\bar{\phi} D^\gamma \phi \right) = -\mathop{\rm Im} \left(\bar{\phi} \partial^\gamma \phi \right) + i \epsilon^{\mu\nu\gamma} \partial_{\mu} B_{\nu} |\phi|^2 \label{L3a} ,\\ \Box \phi =i \epsilon^{\alpha\mu\nu} Q_{\mu\alpha}(B_{\nu},\phi) + Q_0(B_\mu,B^\mu) \phi + Q_{\mu\nu}(B^\mu,B^\nu) \phi\ .\label{L3b} \end{gather} \end{subequations} In this article we shall prove the Theorem stated below which corresponds to exponents $a=\frac14+\epsilon$ and $b=\frac54+\epsilon$. This improves \eqref{H2} by $\frac14-\epsilon$ derivatives in both $a$ and $b$. Compared to \eqref{H1}, it improves $a$ by $\frac12$ derivatives at the expense of $\frac18$ derivatives in $b$. \begin{theorem}\label{lwp} Let $n=2$ and $\frac14 < s < \frac12$. Consider the Cauchy problem for the system \eqref{L3} with initial data in the following Sobolev spaces: \begin{subequations}\label{L4} \begin{gather} B^{\gamma}(0)= b^{\gamma}_{0}\in H^{s+1}(\mathbb{R}^2),\quad \partial_{t} B^{\gamma}(0)= b^{\gamma}_{1} \in H^{s}(\mathbb{R}^2),\\ \phi(0)=\phi_{0}\in H^{s+1}(\mathbb{R}^2),\quad \partial_{t} \phi(0)= \phi_{1} \in H^{s}(\mathbb{R}^2). \end{gather} \end{subequations} Then there exists a $T>0$ and a solution $(B,\phi)$ of \eqref{L3}-\eqref{L4} in $[0,T]\times \mathbb{R}^2$ with $B,\phi \in C^{0}([0,T]; H^{s+1}(\mathbb{R}^2)) \cap C^{1}([0,T]; H^{s}(\mathbb{R}^2)) .$ The solution is unique in a subspace of $C^{0}([0,T]; H^{s+1}(\mathbb{R}^2)) \cap C^{1}([0,T]; H^{s}(\mathbb{R}^2))$, namely in $\mathcal{H}^{s+1,\theta}$, where $\frac34<\theta0$ and let $\theta$ and $\delta$ satisfy \begin{gather*} \frac12 < \theta \leq \min\{1, s+\frac12\} ,\\ 0\leq \delta \leq \min\{1-\theta, s+\frac12-\theta\} . \end{gather*} Then $$\label{NF10} \norm{Q_0(\phi,\psi)}{H^{s,\theta-1+\delta}} \lesssim \norm{\phi}{\mathcal{H}^{s+1,\theta}} \norm{\psi}{\mathcal{H}^{s+1,\theta}}$$ \end{proposition} For a proof of the above proposition, see \cite[estimate (7.5)]{KS}. For $Q=Q_{ij}\, , \, Q_{0j}$ estimate \eqref{NF12} should be compared (if we set $\theta=s+\frac12$ and $\delta=0$) to the following estimate of Zhou \cite{zhou}: $$\label{Ns1} N_{s, s-\frac12}\left( Q_{\alpha\beta}(\phi,\psi) \right) \lesssim N_{s+1, s+\frac12}(\phi) N_{s+1, s+\frac12}(\psi) ,$$ where $\frac14 < s < \frac12$ and $$\label{Ns2} N_{s, \theta}(u)=\norm{w_{+}(\tau,\xi)^s w_{-}(\tau,\xi)^\theta \widetilde{u}(\tau,\xi)}{L^2_{\tau,\xi}} .$$ The spaces in estimate \eqref{NF12} are different, with $\phi$ and $\psi$ slightly less regular in the sense that $\norm{u}{\mathcal{H}^{s,\theta}} \leq N_{s,\theta}(u)$. Moreover we have to account for the extra hyperbolic derivative of order $\delta$ on the left hand side. \begin{proof}[Proof of Proposition \ref{NF11}] We only sketch the proof for $Q=Q_{0j}$. The proof for $Q=Q_{ij}$ is similar. Let \begin{gather*} F(\tau,\xi) = \langle\xi\rangle^{s} w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \widetilde{\phi}(\tau,\xi), \\ G(\tau,\xi) = \langle\xi\rangle^{s} w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \widetilde{\psi}(\tau,\xi) . \end{gather*} Let $H(\tau,\xi)$ be a test function. We may assume $F,G,H\geq 0$. We need to show: \label{0jb} \begin{aligned} & \int\frac{ \langle \xi+\eta \rangle^{s} w_{-}^{\theta-1+\delta} (\tau+\lambda, \xi+\eta) |\tau\eta_{j} - \lambda \xi_{j}| } {\langle\xi\rangle^{s} w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} \\ &\times F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi+\eta)d\tau\, \,d\lambda \,d\xi \,d\eta\\ &\lesssim \norm{F}{L^2} \norm{G}{L^2} \norm{H}{L^2} . \end{aligned} Using $\langle \xi+\eta \rangle^s \leq \langle\xi\rangle^s + \langle \eta \rangle^s$ we see that we need to estimate the following integral (and a symmetric one): $$\label{0jc} \int\frac{ w_{-}^{\theta-1+\delta}(\tau+\lambda, \xi+\eta) |\tau\eta_{j} - \lambda \xi_{j}| F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi+\eta)} { w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+} (\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} \,d\tau \,d\lambda \,d\xi \,d\eta$$ We restrict our attention to the region where $\tau\geq 0$, $\lambda \geq 0$. The proof for all other regions is similar. We use \begin{align*} \tau\eta-\lambda\xi & = (\abs{\xi}\eta-\abs{\eta}\xi) + (\tau-\abs{\xi})\eta - (\lambda-\abs{\eta})\xi \\ & = (\abs{\xi}\eta-\abs{\eta}\xi) + (\abs{\tau}-\abs{\xi})\eta - (\abs{\lambda}-\abs{\eta})\xi \end{align*} to see that, we need to estimate the following three integrals: \begin{gather*} R^{+}=\int\frac{ |\abs{\xi}\eta - \abs{\eta}\xi| F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi+\eta)d\tau d\lambda \,d\xi \,d\eta} { w_{-}^{1-\theta-\delta }(\tau+\lambda, \xi+\eta) w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta) } , \\ T^{+}=\int\frac{ \abs{\abs{\tau}-\abs{\xi}} \abs{\eta} F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi+\eta)d\tau d\lambda \,d\xi \,d\eta} { w_{-}^{1-\theta-\delta }(\tau+\lambda, \xi+\eta) w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} , \\ % L^{+}=\int\frac{\abs{\abs{\lambda}-\abs{\eta}} \abs{\xi} F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi+\eta)d\tau d\lambda \,d\xi \,d\eta} { w_{-}^{1-\theta-\delta}(\tau+\lambda, \xi+\eta) w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} . \end{gather*} We start with $R^{+}$. We have \label{oje} \begin{aligned} &\abs{\abs{\eta} \xi - \abs{\xi} \eta} \\ & \lesssim \abs{\xi}^{1/2} \abs{\eta}^{1/2} \left(\abs{\xi} + \abs{\eta} \right)^{1/2} \left(\abs{|\tau+\lambda|-\abs{\xi+\eta}} + \abs{\abs{\tau}-\abs{\xi}} + \abs{\abs{\lambda}-\abs{\eta}}\right)^{1/2}. \end{aligned} Indeed, \begin{align*} \abs{\abs{\eta}\xi - \abs{\xi}\eta}^2 & = 2 \abs{\eta} \abs{\xi} \left( \abs{\xi} \abs{\eta} - \xi \cdot \eta \right) \\ & = \abs{\eta} \abs{\xi}\left(\abs{\xi}+\abs{\eta} + \abs{\xi+\eta}\right) \left(\abs{\xi}+\abs{\eta} - \abs{\xi+\eta}\right). \end{align*} We have $\abs{\xi}+\abs{\eta} + \abs{\xi+\eta} \leq 2 \left( \abs{\xi} +\abs{\eta}\right)$ and \begin{align*} \abs{\xi}+\abs{\eta} - \abs{\xi+\eta} & = \left( \tau + \lambda - \abs{\xi+\eta}\right) - \left(\lambda - \abs{\eta} \right) - \left(\tau - \abs{\xi} \right) \\ &\leq \abs{ \tau + \lambda - \abs{\xi+\eta}} + \abs{\lambda - \abs{\eta}} + \abs{\tau - \abs{\xi}} , \end{align*} therefore \eqref{oje} follows. Following Zhou \cite{zhou} we use \eqref{oje} to obtain \begin{align*} \abs{\abs{\eta} \xi - \abs{\xi} \eta} &= \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s} \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{1-2s}\\ &\lesssim \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s}\abs{\xi}^{1/2 - s} \abs{\eta}^{1/2 - s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s}\abs{|\tau+\lambda|-\abs{\xi+\eta}}^{1/2 - s} \\ & \quad + \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s}\abs{\xi}^{1/2 - s} \abs{\eta}^{1/2 - s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s}\abs{|\tau|-\abs{\xi}}^{1/2 - s} \\ &\quad + \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s}\abs{\xi}^{1/2 - s} \abs{\eta}^{1/2 - s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s}\abs{|\lambda|-\abs{\eta}}^{1/2 - s} . \end{align*} Therefore, $R^{+} \lesssim R^{+}_{1} + R^{+}_{2} + R^{+}_{3},$ where \begin{align*} R^{+}_{1} &=\int\frac{ \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s}\abs{\xi}^{1/2 - s} \abs{\eta}^{1/2 - s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s} \abs{|\tau+\lambda|-\abs{\xi+\eta}}^{\frac12 - s} } { w_{-}^{1-\theta-\delta}(\tau+\lambda, \xi+\eta) w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} \\ &\quad\times F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi +\eta)\,d\tau \,d\lambda \,d\xi \,d\eta \\ &\leq \int\frac{ \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s} } { w_{-}^{\theta}(\tau,\xi ) w_{-}^{\theta}(\lambda,\eta) |\xi|^{s+1/2} |\eta|^{2s + 1/2} } \\ &\quad \times F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi +\eta)\,d\tau \,d\lambda \,d\xi \,d\eta \end{align*} (we have used the fact that $w_{-}^{s+\frac12-\theta-\delta} (\tau+\lambda,\xi+\eta) \geq 1$. Indeed, $s+\frac12-\theta-\delta > 0$ for small $\delta$, because $\theta < s+\frac12$.) \begin{align*} R^{+}_{2} &=\int\frac{ \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s}\abs{\xi}^{1/2 - s} \abs{\eta}^{1/2 - s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s} \abs{|\tau|-\abs{\xi}}^{1/2 - s}} { w_{-}^{1-\theta-\delta}(\tau+\lambda, \xi+\eta) w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} \\ &\quad\times F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi +\eta)\,d\tau \,d\lambda \,d\xi \,d\eta \\ &\leq \int\frac{ \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s} } { w_{-}^{\theta+s-\frac12}(\tau,\xi ) w_{-}^{\theta}(\lambda,\eta) |\xi|^{s+1/2} \ |\eta|^{2s + 1/2} } \\ &\quad \times F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi +\eta)\,d\tau \,d\lambda \,d\xi \,d\eta \end{align*} (we have used the fact that $w_{-}^{1-\theta-\delta}(\tau+\lambda, \xi+\eta)\geq 1$. Indeed, $1-\theta-\delta \geq 0$ for small $\delta$ because $\theta < s+\frac12 < 1$.) \begin{align*} R^{+}_{3} &=\int\frac{ \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s}\abs{\xi}^{1/2 - s} \abs{\eta}^{1/2 - s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s}\abs{|\lambda| -\abs{\eta}}^{1/2 - s} } { w_{-}^{1-\theta-\delta}(\tau+\lambda, \xi+\eta) w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} \\ &\quad\times F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi +\eta)\,d\tau \,d\lambda \,d\xi \,d\eta \\ &\leq \int\frac{ \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s} } { w_{-}^{\theta}(\tau,\xi ) w_{-}^{\theta+s-\frac12}(\lambda,\eta) |\xi|^{s+1/2} |\eta|^{2s + 1/2} } \\ &\quad \times F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi +\eta)\,d\tau \,d\lambda \,d\xi \,d\eta \end{align*} We present the proof for $R_{2}^{+}$. The proofs for $R_{1}^{+}$ and $R_{3}^{+}$ are similar. We change variables $\tau\mapsto u := |\tau|-\abs{\xi}= \tau-\abs{\xi}$ and $\lambda\mapsto v := |\lambda|-\abs{\eta} = \lambda-\abs{\eta}$ and we use the notation $f_{u}(\xi)=F(u+\abs{\xi},\xi) , \ g_{v}(\eta)=G(v+\abs{\eta},\eta),\ H_{u,v}(\tau',\xi')=H(u+v+\tau',\xi')$ to get \begin{align*} R_{2}^{+} & = \iint \frac{1}{(1+|u|)^{\theta+s-\frac12} (1+|v|)^{\theta} } \Bigl[ \iint \frac{ \abs{\abs{\eta} \xi - \abs{\xi} \eta}^{2s} \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s} } {|\xi|^{s+1/2} |\eta|^{2s + 1/2} } \\ &\quad\times f_{u}(\xi) g_{v}(\eta) H_{u,v}(\abs{\xi} +\abs{\eta},\xi+\eta) \,d\xi \,d\eta \Bigr] \,du \,dv . %\label{0jg} \end{align*} We have $\abs{\abs{\eta} \xi - \abs{\xi} \eta}^2 = 2 \abs{\xi} \abs{\eta} \left(\abs{\xi} \abs{\eta} - \xi\cdot \eta\right)$ therefore \begin{align*} [\cdots] &\lesssim \iint \frac{ \left(\abs{\xi} \abs{\eta} - \xi\cdot \eta\right)^s \left(\abs{\xi} + \abs{\eta} \right)^{1/2 - s}} {|\xi|^{1/2} \ |\eta|^{s + 1/2} } f_{u}(\xi) g_{v}(\eta) H_{u,v}(\abs{\xi} + \abs{\eta},\xi+\eta) d\xi d\eta \\ &\leq \Big(\iint f_{u}(\xi)^2 g_{v}(\eta)^2 d\xi d\eta \Big)^{1/2} K^{1/2} \\ &=\norm{f_{u}}{L^{2}(\mathbb{R}^2)} \norm{g_{v}}{L^{2}(\mathbb{R}^2)} K^{1/2}, \end{align*} where \begin{align*} K&=\iint \frac{ \left(\abs{\xi} \abs{\eta} - \xi\cdot \eta\right)^{2s} \left(\abs{\xi} + \abs{\eta} \right)^{1 - 2s}} {|\xi| \ |\eta|^{2s + 1} } H_{u,v}(\abs{\xi} + \abs{\eta},\xi+\eta)^2 \,d\xi \,d\eta\\ &=\iint \frac{ \left(\abs{\xi'-\eta} \abs{\eta} - (\xi'-\eta)\cdot \eta\right)^{2s} \left(\abs{\xi'-\eta} + \abs{\eta} \right)^{1 - 2s}} {|\xi'-\eta| \ |\eta|^{2s + 1} } \\ &\quad\times H_{u,v}(\abs{\xi'-\eta} + \abs{\eta},\xi')^2 \,d\xi' \,d\eta . \end{align*} We use polar coordinates $\eta = \rho \omega$ to get \begin{align*} K &\lesssim \iiint \frac{ \left(\abs{\xi'-\rho\omega} +\rho - \xi'\cdot \omega\right)^{2s} \left(\abs{\xi'-\rho\omega} + \rho \right)^{1 - 2s}} {|\xi'-\rho\omega| }\\ &\quad\times H_{u,v}(\abs{\xi'-\rho\omega} + \rho ,\xi')^2 \,d\xi' \,d \rho \,d\omega . \end{align*} For fixed $\xi'$ and $\omega$, we change variables $\rho \mapsto \tau' : = \abs{\xi'-\rho\omega} + \rho$ to get \begin{align*} K&\lesssim \iint \Big[\tau'^{1-2s}\ \int_{S^1} \frac{1}{(\tau'-\xi'\cdot\omega) ^{1-2s}}d\omega \Big] H(\tau',\xi')^2 d \xi' d\tau' . \end{align*} From \cite[estimate (3.22)]{zhou} we know that $\tau'^{1-2s}\ \int_{S^1} \frac{1}{(\tau'-\xi'\cdot\omega)^{1-2s}}\, d\omega \lesssim 1 ;$ therefore $K\lesssim \norm{H}{\tilde A}^2$. Putting everything together we get: $R_{2}^{+} \lesssim \Big(\int \frac{ \norm{f_{u}}{L^{2}(\mathbb{R}^2)} }{(1+|u|)^{\theta +s-\frac12} } du \Big) \Big(\int \frac{ \norm{g_{v}}{L^{2}(\mathbb{R}^2)} }{(1+|v|)^{\theta} } dv \Big) \|H\|\,.$ Since $2\theta+2s-1> 2\cdot\frac34 + 2\cdot \frac14 -1 = 1$ and $2\theta > 2\cdot\frac34>1$ we can use the Cauchy-Schwarz inequality to conclude: $R_{2}^{+} \lesssim \norm{\norm{f_{u}}{L^{2}(\mathbb{R}^2)}}{L^{2}_{u}} \norm{\norm{g_{v}}{L^{2}(\mathbb{R}^2)}}{L^{2}_{v}} \|H\| = \|F\| \norm{G}{\tilde A} \norm{H}{\tilde A} .$ This completes the estimates for $R_{2}^{+}$. Next we estimate $T^+$. We use $\abs{\abs{\tau}-\abs{\xi}}\leq w_{+}(\tau,\xi)^{1-\theta} w_{-}(\tau,\xi)^\theta$ to get \begin{align*} T^{+} &=\int\frac{ \abs{\abs{\tau}-\abs{\xi}} \abs{\eta} F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi+\eta)d\tau d\lambda \,d\xi \,d\eta} { w_{-}^{1-\theta-\delta }(\tau+\lambda, \xi+\eta) w_{+}(\tau,\xi) w_{-}^{\theta}(\tau,\xi ) \langle \eta \rangle^{s} w_{+}(\lambda,\eta) w_{-}^{\theta}(\lambda,\eta)} \\ &\leq \int\frac{ F(\tau,\xi) G(\lambda,\eta) H(\tau+\lambda,\xi+\eta)} { \langle \xi \rangle^{\theta} \langle \eta \rangle^{s} w_{-}^{\theta} (\lambda,\eta)} d\tau d\lambda \,d\xi \,d\eta . \end{align*} Changing variables $\tau\mapsto u := |\tau|-\abs{\xi}=\tau-\abs{\xi}$ and $\lambda\mapsto v := |\lambda|-\abs{\eta}=\lambda-\abs{\eta}$ we have \begin{align*} T^{+}& \lesssim \iint \frac{1}{ \langle \xi \rangle^{\theta} \langle \eta \rangle^{s}} \\ & \quad \Big[\iint \frac{F(u+\abs{\xi},\xi) G(v+\abs{\eta},\eta) H(u+v+\abs{\xi}+\abs{\eta},\xi+\eta)}{(1+|v|)^{\theta}} \,du \,dv\Big] \, d \xi \,d\eta. \end{align*} For fixed $\xi$ and $\eta$ we apply \cite[Lemma A]{zhou} in the $(u,v)$-variables to get \begin{align*} T^{+} & \lesssim \iint \frac{1}{ \langle \xi \rangle^{\theta} \langle \eta \rangle^{s}} \norm{F(u+\abs{\xi},\xi)}{L^{2}_u} \norm{G(v+\abs{\eta},\eta)}{L^{2}_v} \\ &\quad\times \norm{H(w+\abs{\xi}+\abs{\eta},\xi+\eta)}{L^{2}_w} d \xi d\eta \\ &= \iint \frac{1}{ \langle \xi \rangle^{\theta} \langle \eta \rangle^{s}} \norm{F(\cdot,\xi)}{L^{2}(\mathbb{R})} \norm{G(\cdot,\eta)}{L^{2}(\mathbb{R})} \norm{H(\cdot,\xi+\eta)}{L^{2}(\mathbb{R})} \,d \xi \,d\eta . \end{align*} Now we do the same in the $(\xi,\eta)$-variables to get \begin{align*} T^+ & \lesssim \norm{\norm{F(\cdot,\xi)}{L^{2}(\mathbb{R})}}{L^{2}_\xi} \norm{\norm{G(\cdot,\eta)}{L^{2}(\mathbb{R})}}{L^2_\eta} \norm{\norm{H(\cdot,\xi')}{L^{2}(\mathbb{R})}}{L^{2}_{\xi'}} \\ &= \norm{F}{\tilde A} \norm{G}{\tilde A} \norm{H}{\tilde A} . \end{align*} The proof for $L^{+}$ is similar. \end{proof} We are also going to need the following product rules' in $H^{s,\theta}$ spaces. \begin{proposition}\label{prod} Let $n=2$. Then $$\label{prod1} \norm{uv}{H^{-c, -\gamma}} \lesssim \norm{u}{H^{a, \alpha}} \norm{v}{H^{b,\beta}},$$ provided that \begin{gather} a+b+c > 1\label{prod2}\\ a+b \geq 0, \quad b+c \geq 0 , \quad a+c \geq 0 \label{prod3}\\ \alpha+\beta+\gamma > 1/2 \label{prod4}\\ \alpha, \beta, \gamma \geq 0 . \label{prod5} \end{gather} \end{proposition} \begin{proof} If $a,b,c\geq 0$, the result is contained in \cite[Proposition A1]{KS}. If not, observe that, due to \eqref{prod3}, at most one of the $a,b,c$ is negative. We deal with the case $c<0$, $a,b\geq 0$. All other cases are similar. Observe that \begin{align*} & \langle\xi\rangle^{-c}\langle |\tau|-|\xi| \rangle^{-\gamma}\abs{\widetilde{uv}(\tau,\xi)} \\ & \lesssim \langle |\tau|-|\xi| \rangle^{-\gamma}\iint \langle \xi-\eta \rangle^{-c}\abs{\widetilde{u}(\tau-\lambda,\xi-\eta)} \abs{\widetilde{v}(\lambda,\eta)} d\lambda d \eta \\ &\quad + \langle |\tau|-|\xi| \rangle^{-\gamma}\iint \abs{\widetilde{u}(\tau-\lambda,\xi-\eta)} \langle \eta \rangle^{-c}\abs{\widetilde{v}(\lambda,\eta)} d\lambda d \eta , \end{align*} therefore \begin{equation*} \norm{uv}{H^{-c,-\gamma}} \lesssim \norm{U v'}{H^{0,-\gamma}} + \norm{u' V}{H^{0,-\gamma}}, \end{equation*} where \begin{gather*} \widetilde{U}(\tau,\xi)=\langle\xi\rangle^{-c}\abs{\widetilde{u}(\tau,\xi)},\\ \widetilde{u'}(\tau,\xi)=\abs{\widetilde{u}(\tau,\xi)},\\ \widetilde{V}(\tau,\xi)=\langle\xi\rangle^{-c}\abs{\widetilde{v}(\tau,\xi)},\\ \widetilde{v'}(\tau,\xi)=\abs{\widetilde{v}(\tau,\xi)}. \end{gather*} Since $a+c\geq 0$, we have \begin{equation*} \norm{Uv'}{H^{0,-\gamma}}\lesssim \norm{U}{H^{a+c,\alpha}} \norm{v'}{H^{b,\beta}} \lesssim \norm{u}{H^{a,\alpha}} \norm{v}{H^{b,\beta}}. \end{equation*} Since $b+c\geq 0$, we have \begin{equation*} \norm{u'V}{H^{0,-\gamma}}\lesssim \norm{u'}{H^{a,\alpha}} \norm{V}{H^{b+c,\beta}} \lesssim \norm{u}{H^{a,\alpha}} \norm{v}{H^{b,\beta}}. \end{equation*} The result follows. \end{proof} \begin{proposition}\label{alg} Let $n=2$. If $s>1$ and $\frac12 < \theta \leq s-\frac12$, then $H^{s,\theta}$ is an algebra. \end{proposition} For the proof of the above proposition, see \cite[Theorem 7.3]{KS}. \begin{proposition} \label{KSThm7.2} Let $n= 2$, $s> 1$, $\frac12 < \theta \leq s - \frac{1}{2}$. Assume that \begin{align} -\theta \leq \alpha \leq 0\, \quad -s \leq a < s+\alpha . \end{align} Then $$\label{KSThm7.2c} H^{a,\alpha}\cdot H^{s,\theta} \hookrightarrow H^{a,\alpha}.$$ \end{proposition} The proof can be found in \cite[Theorem 7.2]{KS}. \subsection*{Proof of Theorem \ref{lwp}} Theorem \ref{lwp} follows by well known methods from the following a-priori estimates (together with the corresponding estimates for differences): For any space-time functions $B, B', \phi, \phi' \in \mathcal{H}^{s+1, \theta}$ and any $\gamma, \mu \in \{0,1,2\}$ we have: \begin{gather} \norm{\phi' \, \partial^\gamma \phi }{H^{s, \theta-1 +\delta}} \lesssim \norm{\phi'}{\mathcal{H}^{s+1,\theta}} \norm{\phi}{\mathcal{H}^{s+1,\theta}} , \label{bound1}\\ \norm{(\partial^\mu B)\, \phi\, \phi' }{H^{s, \theta-1+\delta}} \lesssim \norm{B}{\mathcal{H}^{s+1,\theta}} \norm{\phi}{\mathcal{H}^{s+1,\theta}} \norm{\phi'}{\mathcal{H}^{s+1,\theta}} , \label{bound2}\\ \norm{Q_{\mu\alpha}(B,\phi)}{H^{s, \theta-1 +\delta}} \lesssim \norm{B}{\mathcal{H}^{s+1,\theta}} \norm{\phi}{\mathcal{H}^{s+1,\theta}} , \label{bound3} \\ \norm{Q_{0}(B,B') \phi }{H^{s, \theta-1+\delta}} \lesssim \norm{B}{\mathcal{H}^{s+1,\theta}} \norm{B'}{\mathcal{H}^{s+1,\theta}} \norm{\phi}{\mathcal{H}^{s+1,\theta}} , \label{bound4} \\ \norm{Q_{\mu\nu}(B,B') \phi}{H^{s, \theta-1 +\delta}} \lesssim \norm{B}{\mathcal{H}^{s+1,\theta}} \norm{B'}{\mathcal{H}^{s+1,\theta}} \norm{\phi}{\mathcal{H}^{s+1,\theta}} .\label{bound5} \end{gather} Here \$\frac14