# Reference for singular integral operators such as $(-\Delta)^{-1}$ or $\nabla(-\Delta)^{-1}$

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.

## 1 Answer

Make $$0$$-order operator applying extra $$\partial_j$$ and use your reference.