# Reference request on Pucci extremal operators

While reading [1], I encountered with the concept "Pucci extremal operator" which is defined by: $$M_\Lambda^-(N):=\left(\sum\text{positive eigenvalues of }N\right)+\Lambda\left(\sum\text{negative eigenvalues of }N\right),\text{ and}$$ $$M_\Lambda^+(N):=-M_\Lambda^-(-N),$$ where $$N\in\text{Sym}_{n\times n}$$ and $$\Lambda\geq 1$$.

Then the author claims that the problem $$M_\Lambda^-(D^2u)\leq 0\leq M_\Lambda^+(D^2u)$$ in viscosity sense includes all $$C^2$$ solutions to uniformly elliptic equations of the form $$tr(A(x)D^2u)=0$$ where $$I\leq A(x)\leq\Lambda I$$. Since there is no citation on this, it seems this is well-known result in the field, but I am new to this and I want to know about its motivation, history, etc. I did some article search but I could not find a good reference book or paper that introduces the concept of Pucci extremal operator (of course this may be due to my lacking of searching skill). I will appreciate if anyone would explain the concept or give me some good reference on it. Thank you in advance.

[1] Mooney, Connor, A proof of the Krylov-Safonov theorem without localization, Commun. Partial Differ. Equations 44, No. 8, 681-690 (2019). ZBL1426.35124.

• Incidentally, Connor is active on MathOverflow. If you send him an email pointing out your question, he may drop by and give you an answer. May 6 at 19:54
• @WillieWong That's nice, I may try, thank you for the advice! May 9 at 23:04