Denote $\mathbb{T}$ the 1-torus and let $G : H^1(\mathbb{T}) \to H^1(\mathbb{T})$ be Frechet differentiable such that $G(0) = 0$ and $DG$ maps bounded sets in $H^1$ into bounded sets in $\mathcal{L}(H^1).$ Consider the sequence of operators $\{L_n\}$ defined by $$L_n = \sqrt{n^2-n\Delta}.$$ Assume that there exists a bounded sequence $\{u_n\}$ in $H^1(\mathbb{T})$ such that the sequence $\{L_n u_n\}$ is bounded in $L^2.$
Is the sequence $\{L_n G(u_n)\}$ bounded in $L^2$ ?
