In the 2010 paper https://arxiv.org/pdf/1001.5373.pdf Selberg & Tesfahun prove finite-energy well posedness for the Maxwell-Klein-Gordon system in Lorenz gauge, that is if the equations on $\mathbb{R}^{1+3}$ $$ D^{(A)}_\mu D^{(A)\mu} \phi = m^2 \phi, \qquad \Box A = - \Im(\phi \overline{D^{(A)}\phi} ), \qquad \nabla_\mu A^\mu = 0, $$ where $D^{(A)}_\mu = \nabla_\mu + i A_\mu$, have finite energy data initially, $$ (\phi, \partial_t \phi, \mathbf{E}, \mathbf{B}))|_{t=0} = (\phi_0, \phi_1, \mathbf{E}_0, \mathbf{B}_0) \in H^1(\mathbb{R}^3) \times L^2(\mathbb{R}^3) \times L^2(\mathbb{R}^3) \times L^2(\mathbb{R}^3), $$ then there exists a unique global solution $$ \phi \in C(\mathbb{R};H^1(\mathbb{R}^3)) \cap C^1(\mathbb{R};L^2(\mathbb{R}^3)), \qquad \mathbf{E}, \, \mathbf{B} \in C(\mathbb{R};L^2(\mathbb{R}^3)) $$ (see their Theorem 2.5). Their key observation is that there is null structure in the equation for $\phi$, which when expanded out reads $$ (\Box - m^2)\phi = \mathcal{M}(A,\phi) = 2iA^\mu \partial_\mu \phi + A_\mu A^\mu \phi. $$ They observe this by splitting the spacelike part $\mathbf{A}$ of $A$ into curl-free and divergence-free parts, and then expanding $A^\mu \partial_\mu \phi$ into two terms $P_1$ and $P_2$, which they show are null forms (see section 3.1 in the paper).
Much of their paper is written using wave-Sobolev spaces $H^{s,\theta}$ (see middle of page 12 for a definition), which I am not an expert in and so am finding it hard to extract the details from their paper. My question is simple: does the paper prove that, for a spacetime slab $S_T = [0,T] \times \mathbb{R}^3$, $$ \| \mathcal{M}(A,\phi) \|_{L^2(S_T)} < \infty \quad ?$$ Similarly, do they obtain the estimates for $\mathbf{E}$, $\mathbf{B}$, $$ \Box \mathbf{E} , \quad \Box \mathbf{B} \in H^{-1}(S_T) \quad ? $$ (see middle of page 17 of the paper for relevant statements). Thanks!
