Convergence analysis in convex regularization depending on the smoothness degree of the penalizer

The problem of minimization of the least squares functional with a smooth, lower semi-continuous, convex regularizer $J(\cdot)$ is considered to be solved. Over some compact and convex subset $\Omega$ of the Hilbert space $\mathcal{H},$ the regularizer is implicitly defined as $J(\cdot) : \mathcal{C}^{k}(\Omega , \mathcal{H}) \rightarrow \mathbb{R}_{+}$ where $k \in \{1,2\}.$ So the cost functional associated with some given linear, compact and injective forward operator $\mathcal{T} :\Omega \subset \mathcal{H} \rightarrow \mathcal{H},$ \begin{align} F_{\alpha}(\cdot , f^{\delta}) := \frac{1}{2} \Vert \mathcal{T}( \cdot ) - f^{\delta}\Vert_{\mathcal{H}}^2 + \alpha J(\cdot) , \nonumber \end{align} where $f^{\delta}$ is the given perturbed data with its perturbation amount $\delta$ in it. Convergence of the regularized optimum solution $\varphi_{\alpha(\delta)} \in \mbox{argmin} F_{\alpha}(\varphi , f^{\delta})$ to the true solution $\varphi^{\dagger}$ is analysed depending on the smoothness degree of the regularizer, \textit{i.e.} the cases $k \in \{1,2\}$ in $J(\cdot) : \mathcal{C}^{k}(\Omega , \mathcal{H}) \rightarrow \mathbb{R}_{+}.$ In both cases, we define such a regularization parameter that is in cooperation with the condition \begin{align} \alpha(\delta , f^{\delta}) \in \{ \alpha > 0 \mbox{ }\vert \mbox{ }\Vert\mathcal{T}\varphi_{\alpha}^{\delta} - f^{\delta}\Vert \leq \tau\delta \} , \nonumber \end{align} for some fixed $\tau \geq 1.$ In the case of $k = 2,$ we are able to evaluate the discrepancy $\Vert\mathcal{T}\varphi_{\alpha(\delta)} - f^{\delta}\Vert\leq \tau\delta$ with the Hessian Lipschitz constant $L_H$ of the functional $F_{\alpha}(\cdot , f^{\delta}).$