Transfer of Approximating Test Data from the Sup-Convolution to the Original Function

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation is in force in the dimension $M$, a natural number with $1\le M$. We abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$, and $\tfrac{s}{2}$ denotes the product of $s\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field. The set $\mathbb{R}^{M}$ is open, directly from Open Subset of a Metric Space. That a quadruple is approximable by test data from above for a function on an open set is as defined there.

Let $v:\mathbb{R}^{M}\to\mathbb{R}$ be upper semicontinuous on $\mathbb{R}^{M}$, let $C\in\mathbb{R}$ be an upper bound for the set of values of $v$, let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$, so that $\lambda^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, and let $v^{\lambda}$ be the sup-convolution of $v$ with parameter $\lambda$.

Let $\eta_{0},q_{0}\in\mathbb{R}^{M}$ and $Y\in\mathcal{S}(M)$, and suppose that the quadruple $\bigl(\eta_{0},v^{\lambda}(\eta_{0}),q_{0},Y\bigr)$ is approximable by test data from above for $v^{\lambda}$, the domain being $\mathbb{R}^{M}$. Put

$$z_{0}=\eta_{0}+\lambda^{-1}q_{0}.$$

Then the following hold.

1. (Value identity) (i) $v(z_{0})=v^{\lambda}(\eta_{0})+\tfrac{1}{2}\,\lambda^{-1}\lVert q_{0}\rVert^{2}$.

2. (Transfer of the test data) (ii) The quadruple $\bigl(z_{0},v(z_{0}),q_{0},Y\bigr)$ is approximable by test data from above for $v$, the domain being $\mathbb{R}^{M}$.