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**](http://dx.doi.org/10.1080/03605302.2019.1581807), Commun. Partial Differ. Equations 44, No. 8, 681-690 (2019). [ZBL1426.35124](https://zbmath.org/?q=an:1426.35124).



Reference request on Pucci extremal operators
The original references are the works of Pucci [2] and [3] which, however, are written in Italian. A perhaps more accessible introduction to these kind of operators is found in the monograph [1], §2.2, pp. 14-17, by Caffarelli and Cabré: in the latter reference, particularly relevant to your questions are lemma 2.12, §2.2 pp. 15-16 and the final part of remark 2, §2.2 p. 16.


**References**


[1] Luis A. Caffarelli, Xavier Cabré, *Fully nonlinear elliptic equations*, (English)
Colloquium Publications. 43. Providence, RI: American Mathematical Society (AMS). v, 104 p. (1995), [DOI: 10.1090/coll/043](http://dx.doi.org/10.1090/coll/043), [MR1351007](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1351007), [Zbl 0834.35002](https://www.zbmath.org/?q=an%3A0834.35002).


[2] Carlo Pucci, "Operatori ellittici estremanti", (in Italian) Annali di Matematica Pura e Applicata (4) 72 (1966), 141--170, [DOI: BF02414332](https://doi.org/10.1007/BF02414332), [MR0208150](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0208150), [Zbl 0154.12402](https://www.zbmath.org/?q=an%3A0154.12402).


[3] Carlo Pucci, "Operatori massimanti" (in Italian) *Monge-Ampère equations and related topics. Proceedings of a Seminar held in Firenze, September - October 1980*, Roma: Istituto Nazionale di Alta Matematica ”Francesco Severi”, 169-176 (1982), [Zbl 0519.35076](https://www.zbmath.org/?q=an%3A0519.35076).


