This is my first post in two years (some other day I can ponder again about why I keep failing at blogging regularly). Perhaps unsurprisingly, two years later I still find myself thinking about the structure of nonlocal operators! What I want to discuss here is a very basic observation that I came across while working on my latest paper with Russell Schwab (where we revisit our 2016 preprint on min-max formulas for nonlocal operators). As such, this post can be seen as a kind of follow up to this one, where the Global Comparison Property and other basic definitions are discussed.
The idea is how to describe elliptic integro-differential equations in a way that is entirely analogous to
\[F(D^2u,\nabla u(x),u(x),x) = 0,\]which is how local non-divergence equations are often considered. That is, the operator in the equation depends on the Hessian, gradient, and value of the function at $x$, as well as the location $x$ itself. To put it differently, a local fully nonlinear equation is described by a real valued function
\[F: \text{Sym}(\mathbb{R}^d) \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\]which is monotone with respect to its $\text{Sym}(\mathbb{R}^d)$ and $\mathbb{R}$ arguments –here $\text{Sym}(\mathbb{R}^d)$ denotes the space of symmetric real matrices of dimension $d$.
The way to achieve a similar description for nonlocal operators involves a functional following space, which arises naturally when dealing with elliptic integro-differential operators. The space is defined by
\[L^\infty_\beta = \{ h \in L^\infty(\mathbb{R}^d) \text{ such that } |h(y)| = O(|y|^\beta) \text{ as } |y| \to 0 \},\]when $\beta \in (1,2)$ –the discussion below can be extended to all $\beta \in [0,2]$, but we focus on this range for the sake of a simplified presentation. The space $L^\infty_\beta$ has a topology given by the norm
\[\| h \|_{L^\infty_\beta} := \sup \limits_{y \in \mathbb{R}^d} |h(y)| (\min\{1,|y|^{\beta} \})^{-1}.\]Now, suppose we are given a continuous, real valued function
\[F:L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\]which is monotone increasing with respect to its $L^\infty_\beta$ and $\mathbb{R}$ arguments. To such a function $F$ we can associate an operator $I = I_F$, as follows
\[I(u,x) := F(\delta_x u,\nabla u(x),u(x),x),\]here, $\delta_x u$ is the function in $L^\infty_\beta$ defined by
\[\delta_x u(y) := u(x+y)-u(x)-\chi_{B_1}(y)\nabla u(x)\cdot y.\]It follows, thanks to the monotonicity assumed on $F$, that $I_F$ satisfies the Global Comparison Property (GCP).
All operators with the GCP arise in this fashion. This can be seen easily, in fact. First, note that given
\[(h,p,z,x_0) \in L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d,\]we can define a function $u_{h,p,z,x_0}:\mathbb{R}^d\to \mathbb{R}$ by
\[u_{h,p,z,x_0}(x) = z+\chi_{B_1}(x-x_0)p\cdot (x-x_0)+ h(x-x_0),\;\;\forall\;x\in\mathbb{R}^d.\]This function may not be of class $C^\beta$ in all of $\mathbb{R}^d$, but it has enough regularity at $x_0$ for our purposes.
Now with this definition at hand, suppose $I:C^\beta(\mathbb{R}^d)\to C^0(\mathbb{R}^d)$ is an operator satisfying the GCP, then we may define $ F:L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$ via
\[F(h,p,z,x) := I( u_{h,p,z,x},x).\]Since $u_{h,p,z,x}$ is sufficiently regular, the operator is clasically defined for $u_{h,p,z,x}$ at the point $x$ be clasically defined. Furthermore, by construction, we have
\[u_{h,p,z,x_0}(x_0) = z,\; \nabla u_{h,p,z,x_0}(x_0) = p,\; \delta_{x_0} u_{h,p,z,x_0} = h,\]so it follows that for $F$ thus constructed we have $I = I_F$, and that $F$ is monotone increasing with respect to $h$ and $z$ as long as $I$ satisfies the GCP.
Therefore, when thinking of a nondivergence integro-differential equation, a good way to think about it is as a function
\[F:L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\]As one last remark: note that the (positive) elements in the dual of $L^\infty_\beta$ are simply the Levy measures that are integrable against
\[\min\{1,|y|^\beta\}.\]