@DeaneYang What "section" are you referencing? Also, can you give an explicit example? Maybe a worked parametrix with specific coefficients?

Thanks, everyone. It's nice to know that I wasn't just missing something trivial. This is a good theorem to know!

The value $p=4$ might not be special, but it is the only instance in which we are able to prove the $\limsup$ convergence explicitly.

Oh, of course! Sorry. Thanks again.

Thanks. That really helps. This is the [Hölder interpolation][1] theorem you're referencing, correct? In the case when $p_t\in(4,6)$ though, we end up with $$\lambda_k^{-\delta(p_t)}||e_{\lambda_k}||_{p_t} \leq \left( \lambda_k^{-\delta(4)}||e_{\lambda_k}||_4 \right)^{1-t}\left( \lambda_k^{-\delta(6)}||e_{\lambda_k}||_6 \right)^{t}$$ and we can no longer rely on the unit-norm in $L^{2}(M)$ nor the fact that $\delta(2)=0$. How do we proceed from here? [1]: en.wikipedia.org/wiki/…