I posted this on Math StackExchange, but I figured it couldn't hurt to ask here as well.


I'm trying to decipher a particular claim in a paper I'm reading, but I just can't seem to figure it out.


The **M. Riesz Interpolation Theorem** says:



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> Let $T:L^{p\_0}\cap L^{p\_1} \to L^{q\_0} \cap L^{q\_1}$ be a linear map satisfying: $$||Tf||\_{q\_j}\leq M\_{j}||f||\_{p\_j}, \quad j=0,1$$ with $1\leq p\_j,q\_j\leq\infty$. Then, if for $0< t <1$, we define $$\frac{1}{p\_t}=\frac{1-t}{p\_0}+\frac{t}{p\_1}, \quad \frac{1}{q\_t}=\frac{1-t}{q\_0}+\frac{t}{q\_1},$$ we have $$||Tf||\_{q\_t}\leq (M\_0)^{1-t}(M\_1)^{t}||f||\_{p\_t}, \quad f\in L^{p\_0}\cap L^{p\_1}.$$
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Note that $p\_0 \leq p\_t \leq p\_1$ and $q\_0 \leq q\_t \leq q\_1$.


The paper I'm reading gives the following:



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> $$||e\_\lambda||\_p \leq C \lambda^{\delta(p)}||e\_\lambda||\_{2}, \quad 2\leq p \leq 6$$
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> $$\delta(p) = \frac{1}{2}\left(\frac{1}{2} - \frac{1}{p} \right)$$
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Here $e\_\lambda$ is an eigenfunction (with eigenvalue $\lambda$) of $\sqrt{-\Delta}$ on a fixed, compact Riemannian manifold $M$, with no boundary, and $C$ is a fixed constant depending only on the manifold. Note in particular that $\delta(2)=0$ and $\delta(6)=\tfrac{1}{6}$.


It then proceeds to claim that using the above result for $p=2$ and $p=6$, along with the interpolation theorem, we have the following:



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> $$\limsup\_{k\to\infty} \lambda\_{k}^{-\delta(p)}||e\_{\lambda\_k}||\_{L^{p}(M)} = 0$$ is true for $2<p<6$ *if and only if* it is true for $p=4$, where the eigenfunctions are all taken to have unit $L^{2}(M)$-norms, and $\lambda\_1 \leq \lambda\_2 \leq \cdots.$
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If we let $T=E\_{k}$ be the projection onto the $k^{\text{th}}$ eigenspace, that is,
$$E\_{k}f = \langle f,e\_{\lambda\_k}\rangle e\_{\lambda\_k}(x), \quad \langle f,e\_{\lambda\_k}\rangle = \int\_{M}f(x)\overline{e\_{\lambda\_k}(x)}dx, \quad \forall f\in L^{2}(M)$$
then $E\_{k}:L^{2} \to L^{q\_0}\cap L^{q\_1}$, for any $2\leq q\_0,q\_1\leq 6$, and it seems like we should be able to use the interpolation theorem in some manner.


I'm lost though when it comes to showing the above claim is true for $2<p<6$, assuming it is true for $p=4$. Moreover, I feel like I've seen this kind of trick a lot, where we only have to prove a statement for some particular value of $p$ in order for it to be true for some larger range of values. What am I missing?



using the M. Riesz Interpolation Theorem
My answer is for $p \in (2,4)$, the other case should follow similar.


Let $t \in (0,1)$ be given, such that $$\frac1p = \frac{1-t}{2}+\frac{t}{4}.$$
I will go to use the following consequence of Hölder (this is similar to the Riesz Interpolation Theorem, but somewhat simpler to use in your case):
$$\lVert f \rVert\_p \le \lVert f\rVert\_2^{1-t} \lVert f\rVert\_4^t$$
Using $\delta(p) = (1-t)\;\delta(2) + t \; \delta(4)$ you find
$$\lambda\_k^{-\delta(p)} \lVert e\_k \rVert\_p \le \Big(\lambda\_k^{-\delta(2)}\lVert e\_k\rVert\_2\Big)^{1-t}\Big(\lambda\_k^{-\delta(4)}\lVert e\_k\rVert\_4\Big)^t = \Big(\lambda\_k^{-\delta(4)}\lVert e\_k\rVert\_4\Big)^t.$$
This shows, that your lim-sup holds for all $p \in (2,4)$ if it holds for $p = 4$.


**Edit:** A intuition for the statement is the following:
You have $$\lambda\_k^{-\delta(p)} \lVert e\_k \rVert\_p$$ is bounded for $p \in [2,6]$ as $k \to \infty$. If for some $p \in (2,6)$, the $\limsup$ is zero, than you can interpolate and obtain a zero $\limsup$ for all $p \in (2,6)$. Note there is nothing special about $p = 4$.


A similar argument is: if a sequence of functions $f\_k$ is bounded in $L^p$ and converges towards zero in $L^q$, $q < p$, then it converges to $0$ in $L^r$ for all $r \in [q,p)$.


