The reason I'm interested in this is that I have a curvature tensor $F$ whose $L^p$ norm blows up at finitely many points $Z$ of a compact manifold $X$, but I have good local estimates for the $L^p$ norm of $F$ away from these points. However, I would like to have a bound on the $L^p$ norm of $F$ over the entire $X-Z$. What would be your strategy to obtain this bound? I don't have extra assumptions on $F$, and an $L^{\infty}$ bound seems too strong.

No problem- thank you for the comments!

Why? $M$ is supposed to have finite volume, and the constant $C_{M,H}$ depends on both $M$ and $H$.

Even for a bounded domain $U\subset\mathbb{R}^n$, and a function $f$ defined on $U$, the answer doesn't seem obvious to me. For constant functions $f=K$, the statement is true- the local estimate gives a bound on $K$, so the global estimate is given by a constant depending on $K$ and the volume of $U$.

Ah, yes! I changed the power (2q-n) to (q-n). Thanks!