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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/…