The Theorem on Sums for a Global Quadratic Bound, in Two Groups of Variables

Throughout we work in the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which it rests, is in force in the dimensions $m$, $n$ and $N=m+n$, for natural numbers $m$ and $n$ with $1\le m$ and $1\le n$. In particular $\iota:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{N}$ is the concatenation map, a bijection. We use in addition the square $A^{2}=AA$ of a square real matrix and the notation $-aI_{N}$ for $(-a)I_{N}$, an element of $\mathcal{S}(N)$ for every real $a$; and for $X_{1}\in\mathcal{S}(m)$ and $X_{2}\in\mathcal{S}(n)$, $X_{1}\oplus X_{2}\in\mathcal{S}(N)$ is the block diagonal matrix of Block Diagonal Symmetric Matrices and the Blocks of a Symmetric Matrix (i).

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 sets $\mathbb{R}^{m}$, $\mathbb{R}^{n}$ and $\mathbb{R}^{N}$ are 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 $u_{1}:\mathbb{R}^{m}\to\mathbb{R}$ be upper semicontinuous on $\mathbb{R}^{m}$ and let $u_{2}:\mathbb{R}^{n}\to\mathbb{R}$ be upper semicontinuous on $\mathbb{R}^{n}$; assume that the sets of values of $u_{1}$ and of $u_{2}$ have upper bounds $C_{1}$ and $C_{2}$ in $\mathbb{R}$, and that

$$u_{1}\bigl(0_{\mathbb{R}^{m}}\bigr)=0,\qquad u_{2}\bigl(0_{\mathbb{R}^{n}}\bigr)=0 .$$

Let $A\in\mathcal{S}(N)$ and assume the global quadratic bound

$$u_{1}(\xi)+u_{2}(\eta)\ \le\ \tfrac{1}{2}\,\iota(\xi,\eta)\cdot\bigl(A\,\iota(\xi,\eta)\bigr)\qquad\text{for all }\xi\in\mathbb{R}^{m}\text{ and }\eta\in\mathbb{R}^{n}.$$

Let $\varepsilon\in\mathbb{R}$ be positive, so that $\varepsilon^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, and put $B=A+\varepsilon A^{2}$, which lies in $\mathcal{S}(N)$ by A Weighted Young Inequality and the Splitting of a Quadratic Form (ii).

Then there exist $X_{1}\in\mathcal{S}(m)$ and $X_{2}\in\mathcal{S}(n)$ for which both of the following hold.

1. (Test data at the origin) (i) The quadruple $\bigl(0_{\mathbb{R}^{m}},u_{1}(0_{\mathbb{R}^{m}}),0_{\mathbb{R}^{m}},X_{1}\bigr)$ is approximable by test data from above for $u_{1}$, the domain being $\mathbb{R}^{m}$, and the quadruple $\bigl(0_{\mathbb{R}^{n}},u_{2}(0_{\mathbb{R}^{n}}),0_{\mathbb{R}^{n}},X_{2}\bigr)$ is approximable by test data from above for $u_{2}$, the domain being $\mathbb{R}^{n}$.

2. (Two-sided matrix bound) (ii)

$$-\bigl(\varepsilon^{-1}+\lVert A\rVert\bigr)I_{N}\ \preceq\ X_{1}\oplus X_{2}\ \preceq\ A+\varepsilon A^{2}.$$