Let $(M,g)$ be a Riemannian manifold. Let $E=(E_1,\dots,E_n)$ be an orthonormal frame for $M$.
So for $M$ itself we have a natural $C^k$-norm $\|f\|_{C^k_g(M)}:=\max\limits_{1\le m\le k}\sup\limits_{x\in M}|\nabla^mf(x)|$ where $|\nabla^mf(x)|=\max\limits_{v_i\in T_xM,|v_i|=1}\langle\nabla^m f(x),v_1\otimes\dots v_m\rangle_g$.
On the other hand we can define norm structures along the frame $E$ by viewing $E_j$ as differential operator $\|f\|_{C^k_E(M)}:=\max\limits_{1\le m\le k}\{\|E_{i_1}\dots E_{i_m}f\|_{C^0(M)}:1\le i_j\le n\}$.
My question is: Is there a universal constant $C=C(n)>0$ such that $\|f\|_{C^2_g(M)}\le C\|f\|_{C^2_E(M)}$ for all $(M,g,E)$ and $f$?
More specifically, can we have $\|(\nabla_{E_i}E_j)f\|_{C^0}\lesssim_n\|E_iE_jf\|_{C^0}+\|E_jE_if\|_{C^0}$?
Note that for $C^1$, $|\nabla f(x)|=\max\limits_{v\in\mathbb R^n,|v|=1}df_x(v_1E_1+\dots+v_nE_n)=\sqrt{\sum_{i=1}^n|E_if(x)|^2}$. So the analogically control holds with $C_1=1$.
More generally, does the analogy holds for $C^k$?
