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Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant. Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > 0 \right\}$.

We say that a continuous function $u: \mathbb{C}^n \rightarrow \mathbb{R}$ is a viscosity $\Gamma$-subsolution (resp. viscosity $\Gamma$-subsolution) if $\forall h \in C^2$, $u-h$ achieves local maximum (resp. local minimum) at $z_0$, we have $\lambda(h_{i \bar{j}}(z_0)) \in \overline{\Gamma}$ (resp. $\lambda(h_{i \bar{j}}(z_0)) \in \mathbb{R}^n \backslash \Gamma$), where $\lambda(A)$ is the $n$-tuple of eigenvalues of the hermitian matrix $A$.

In addition, denote the standard convolution of $u$ as follows $$[u]_r(z) := \int_{\mathbb{C}^n} u(z+r\tilde{z}) \eta(\tilde{z}) \Omega(\tilde{z}) = \int_{\mathbb{C}^n} u(z') \eta\left(\frac{z'-z}{r}\right) \frac{1}{r^{2n}} \Omega(z'),$$ where $\Omega$ is the standard wolume form on $\mathbb{C}^n$, and $\eta$ is a standard mollifier with $\eta \geq 0$, $\text{supp}(\eta) \subset B_1(0)$, and $\int_{\mathbb{C}^n} \eta(z') \Omega(z') = 1$.

Now, let a continuous function $u$ be a viscosity $\Gamma$-solution (ie. viscosity $\Gamma$-subsolution and viscosity $\Gamma$-supersolution). Is $[u]_r$ still a viscosity $\Gamma$-subsolution?

I only understand that the statement holds for $C^2$ viscosity $\Gamma$-subsolution. Since we have $\lambda(u_{i\bar{j}}(z)) \in \overline{\Gamma}$ for all $z$, this indicates that $$\lambda(([u]_r)_{i\bar{j}}) = \int_{\mathbb{C}^n} \lambda(u_{i\bar{j}}(z+r\tilde{z})) \eta(\tilde{z}) \Omega(\tilde{z}) \in \overline{\Gamma}$$ by the convexity of $\Gamma$.

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