The following fundamental property of subharmonic functions was proved by S.A.Imomkulov [4] in 1992:

Theorem. Let $f(x)$ be a subharmonic function on a domain $D\subset\mathbb{R}^n$. Then for any $\varepsilon>0$ there is $g\in C^2(\mathbb{R}^n)$ such that $ |\{x\in D:\ f(x)\neq g(x)\}|<\varepsilon. $

Unfortunately, the result is not known. According to MathSciNet the paper of Imomkulov has zero citations. Recently, another proof has been obtained in [2] although the authors were not aware of the result of Imomkulov.

Convex functions are subharmonic and in that special case the above result was proved in [1] and in [3]. Note that both of the papers were published after the paper of Imomkulov.

[1] G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differential Equations 2 (1994), 17–27.

[2] G. Alberti, S. Bianchini, C. G. Stefano; Crippa, On the $L^p$-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30 (2014), no. 1, 349–367.

[3] L. C. Evans, W. Gangbo, W. Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999), no. 653

[4] S. A. Imomkulov, Twice differentiability of subharmonic functions. (Russian. Russian summary) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 877--888; translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 157–167

The Lusin theorem says that a measurable function coincides with a continuous function away from a set of measure less than $\varepsilon$. A result of Federer says that an a.e. differentiable function coincides with a $C^1$ function away from a set of measure less than $\varepsilon$. So such functions have a $C^1$ Lusin property. Imomkulov proved an analogous $C^2$ Lusin property for subharmonic functions.

The following fundamental property of subharmonic functions was proved by S.A.Imomkulov [4] in 1992:

Theorem. Let $f(x)$ be a subharmonic function on a domain $D\subset\mathbb{R}^n$. Then for any $\varepsilon>0$ there is $g\in C^2(\mathbb{R}^n)$ such that $ |\{x\in D:\ f(x)\neq g(x)\}|<\varepsilon. $

Unfortunately, the result is not known. According to MathSciNet the paper of Imomkulov has zero citations. Recently, another proof has been obtained in [2] although the authors were not aware of the result of Imomkulov.

Convex functions are subharmonic and in that special case the above result was proved in [1] and in [3]. Note that both of the papers were published after the paper of Imomkulov.

[1] G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differential Equations 2 (1994), 17–27.

[2] G. Alberti, S. Bianchini, C. G. Stefano; Crippa, On the $L^p$-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30 (2014), no. 1, 349–367.

[3] L. C. Evans, W. Gangbo, W. Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999), no. 653

[4] S. A. Imomkulov, Twice differentiability of subharmonic functions. (Russian. Russian summary) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 877--888; translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 157–167

The following important and almost unknown result of S.A.Imomkulov [4] was proved in 1992:

Theorem. Let $f(x)$ be a subharmonic function on a domain $D\subset\mathbb{R}^n$. Then for any $\varepsilon>0$ there is $g\in C^2(\mathbb{R}^n)$ such that $ |\{x\in D:\ f(x)\neq g(x)\}|<\varepsilon. $

In my opinion this is a result of fundamental importance, but according to MathSciNet the paper of Imomkulov has zero citations. Recently another proof has been obtained in [2] although the authors were not aware of the result of Imomkulov.

Convex functions are subharmonic and in that special case the above result was proved in [1] and in [3]. Note that both of the papers were published after the paper of Imomkulov.

[1] G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differential Equations 2 (1994), 17–27.

[2] G. Alberti, S. Bianchini, C. G. Stefano; Crippa, On the $L^p$-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30 (2014), no. 1, 349–367.

[3] L. C. Evans, W. Gangbo, W. Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999), no. 653

[4] S. A. Imomkulov, Twice differentiability of subharmonic functions. (Russian. Russian summary) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 877--888; translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 157–167

The following fundamental property of subharmonic functions was proved by S.A.Imomkulov [4] in 1992:

Theorem. Let $f(x)$ be a subharmonic function on a domain $D\subset\mathbb{R}^n$. Then for any $\varepsilon>0$ there is $g\in C^2(\mathbb{R}^n)$ such that $ |\{x\in D:\ f(x)\neq g(x)\}|<\varepsilon. $

Unfortunately, the result is not known. According to MathSciNet the paper of Imomkulov has zero citations. Recently, another proof has been obtained in [2] although the authors were not aware of the result of Imomkulov.

Convex functions are subharmonic and in that special case the above result was proved in [1] and in [3]. Note that both of the papers were published after the paper of Imomkulov.

[1] G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differential Equations 2 (1994), 17–27.

[2] G. Alberti, S. Bianchini, C. G. Stefano; Crippa, On the $L^p$-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30 (2014), no. 1, 349–367.

[3] L. C. Evans, W. Gangbo, W. Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999), no. 653

[4] S. A. Imomkulov, Twice differentiability of subharmonic functions. (Russian. Russian summary) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 877--888; translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 157–167