I'm currently working on understanding certain mean-field limits in kinetic theory, and the equations I'm working with are usually of the form $$\partial_t f +v\cdot\nabla_x f \pm c\nabla(-\Delta)^{-1}\rho_f\cdot\nabla_v f=0$$ for $f:\mathbb R_x^d\times\mathbb R_v^d\times\mathbb R_t\to\mathbb R$ and where $\rho_f = \int\! f\,\mathrm dv$, or $$\partial_t f + A\nabla(-\Delta)^{-1}f\cdot\nabla f=0$$ for $f:\mathbb R_x^d\times\mathbb R_t\to\mathbb R$, for some matrix $A:\mathbb R^d\to\mathbb R^d$.
What all of these equations have in common is that they are integro-differential equations, where we must understand the boundedness properties of the operator $\nabla(-\Delta)^{-1}$, typically expressible on a suitable domain $\Omega$ using Green's functions as $$ \mathscr K u(x):=\nabla(-\Delta)^{-1}u(x)=\int_\Omega\!\nabla_x G_\Omega(x,y)u(y)\,\mathrm dy. $$ Now, when $d=2$, I know that we have $\mathscr K:L^1\cap L^\infty(\Omega)\to C^{1,\mathrm{log}}(\Omega)$, where $C^{1,\log}(\Omega)$ denotes the log-Lipschitz functions. This fact is used to develop the 2D well-posedness theory of the incompressible Euler equation.
Are there any similar results and references that can be used generally for SIOs like $\mathscr K=\nabla(-\Delta)^{-1}$? I suspect some tools from harmonic analysis and PDO theory will come in handy, but I don't know of any good reference for integrating SIOs – the main reference I know, Christ's Lectures on Singular Integral Operators deals mainly with degree $0$ operators.
