Suppose we have a compact Riemmanian manifold $(M^n,g)$ without boundary, and some class $\{g_u\}$ of conformal metrics $g_u:=e^{2u}g$, where $\{u\}\subset C^\infty(M^n)$. For some $k,p$ suppose there exists a constant $C_1=C_1(g)$ such that each $u$ in this class has corresponding $W^{k,p}(M^n,g_u)$-norm bounded by $C_1$: \begin{equation} ||u||_{W^{k,p}(M^n,g_u)} \leq C_1. \end{equation}
Can we conclude there exists $C_2=C_2(g)$ such that each $w$ in this class has $W^{k,p}(M^n,g)$-norm bounded by $C_2$, i.e. \begin{equation} ||u||_{W^{k,p}(M^n,g)} \leq C_2? \end{equation}
UPDATE: The only thing I can think of is if $p$ is high enough then from Sobolev embeddings we get a bound on the Holder norms, but is this uniform in $g$? I should think uniform bounds on the Sobolev norms does imply uniform bounds on the Holder norms. If this is the case, then in particular we get uniform bounds on all the functions $u$, in which case just doing conformal transformations in the integrals (see the comments) answers in the affirmative.