Penalty-Subordinate Growth of a Function on the Penalty Domain

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $u:\mathcal{D}\to\mathbb{R}$.

1. (Growth from above)(i) The function $u$ has penalty-subordinate growth from above if for every positive $\delta\in\mathbb{R}$ there is $C\in\mathbb{R}$ such that

$$u(\mu)\le C+\delta\,\mathcal{E}(\mu)\qquad\text{for every }\mu\in\mathcal{D}.$$

2. (Growth from below)(ii) The function $u$ has penalty-subordinate growth from below if for every positive $\delta\in\mathbb{R}$ there is $C\in\mathbb{R}$ such that

$$-C-\delta\,\mathcal{E}(\mu)\le u(\mu)\qquad\text{for every }\mu\in\mathcal{D}.$$