Extremal operators

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The extremal operator associated to some class of linear operators $\mathcal{L}$ represent the maximal and minimal value that $Lu(x)$ can take from all possible choices of $L \in \mathcal L$.

The extremal operators are used to define uniform ellipticity for nonlocal operators. In fact, the extremal operators are also the maximal and minimal nonlinear uniformly elliptic operators with respect to $\mathcal L$ that vanish at zero.

Given any family of linear integro-differential operators $\mathcal{L}$, we define the extremal operators $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$: \begin{align*} M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} \, L u(x) \\ M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} \, L u(x) \end{align*}

If $\mathcal L$ consists of purely second order operators of the form $Lu = \mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators, which have the formula \begin{align*} P^+(D^2 u) &= \Lambda \ \mathrm{tr}(D^2u^+) - \lambda \ \mathrm{tr}(D^2u^-)\\ P^-(D^2 u) &= \lambda \ \mathrm{tr}(D^2u^+) - \Lambda \ \mathrm{tr}(D^2u^-) \end{align*}

If $\mathcal{L}$ consists of all symmetric purely integro-differential operators, uniformly elliptic of order $s$, then the extremal operators have the formula[1] \begin{align*} M^+\, u &= \int_{\R^n} \left( \Lambda \delta u(x,y)^+ - \lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y \\ M^-\, u &= \int_{\R^n} \left( \lambda \delta u(x,y)^+ - \Lambda \delta u(x,y)^- \right) \frac{(2-s)}{|y|^{n+s}} \mathrm d y \end{align*} where $\delta u(x,y) = (u(x+y) + u(x-y) - 2u(x))$. These two extremal operator are sometimes called "the monster Pucci operators" (even though there are other operators that are easily more "monstruous")