The Cut-Off Shifted Operators of the Wick-Square Problem: the Hypotheses of the Convergence Theorem with Constants Independent of the Cutoff, Convergence on Bounded Test Data, and Compactness of the Triple

In the setting of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation, let $C_{g},\ell_{g}\in\mathbb{R}$ be nonnegative and let $g:V\to\mathbb{R}$ satisfy $|g(x)|\le C_{g}$ and $|g(x)-g(y)|\le\ell_{g}|x-y|_{H}$ for all $x,y\in V$. Let $F^{\sharp}$ be the shifted operator for $g$ and, for $N\in\mathbb{N}$, let $\Pi_{N}$, $f^{N}$, $G_{N}$, $B^{N}_{q}$ be the cut-off data and $F^{\sharp}_{N}$ the cut-off shifted operator for $g$. Then the following hold.

1. (Uniform hypotheses)¶ For every $N\in\mathbb{N}$: $G_{N}\in\mathrm{Sym}(H)$ with $0_{\mathrm{Sym}}\preceq G_{N}\preceq I_{H}$; $B^{N}_{q}$ is a monotone nonlinearity for $(H,V,A)$; and $|g(\Pi_{N}x)|\le C_{g}$ and $|g(\Pi_{N}x)-g(\Pi_{N}y)|\le\ell_{g}|x-y|_{H}$ for all $x,y\in V$.

2. (Convergence of the operators)¶ The sequence $(F^{\sharp}_{N})_{N\in\mathbb{N}}$ converges to $F^{\sharp}$ on bounded test data.

3. (Compactness)¶ Every sequence in $V$ that is bounded in $V$ has a subsequence that converges in $H$.