Constants in Meyer inequalities

Question

Let us first recall Meyer inequality in the Malliavin calculus framework $$\|\delta(u)\|_{L^p}\leq C_p\|u\|_{\mathbb{D}^{1,p}},\qquad \forall u\in \mathbb{D}^{1,p},$$ where $\delta$ is the skorohod integral, $\mathbb{D}^{1,p}$ is the Sobolev space, and $\|\cdot\|_{\mathbb{D}^{1,p}}$ is given by: $$\|u\|_{\mathbb{D}^{1,p}}=\mathbb{E}[\|u\|^p_H]+\mathbb{E}[\|Du\|^p_{H\otimes H}],$$ here $D$ denotes the Malliavin derivatives. I want to estimate the constants $C_p$ for $p$ large enough, i.e., I want some thing like $C_p=O(p^{\beta})$.