Inequality in $L^2$ implies inequality in $L^p$ For a project I'm working on, I have an inequality that applies for all functions in $L^2$ but I would like to apply it for all functions in $L^p$ instead ($p>2$). Specifically, my setup is as follows:
Let $X$ be a $\sigma$-finite measure space (I'd like to apply this in the case that $X$ is a manifold). Suppose that $T: L^p(X)\to L^p(X)$ for any $2\leq p<\infty$ is sublinear and that
$$
\|u\|_{L^2}\leq \|Tu\|_{L^2}
$$
for all $u\in L^2$. I would like to conclude that given any other $2<p<\infty$, that we have
$$
\|u\|_{L^p}\leq C_p\|Tu\|_{L^p}
$$
for all $u\in L^p$ where $C_p$ can depend on $p$.
 A: I'm posting an answer to my own question for anyone else who may come across this in the future.
This seems to be false in general, but I have posted an updated question here, in which I ask if the result holds if you also know that
$$
\|u\|_\infty\leq C_\infty \|Tu\|_\infty.
$$
In other words, can you apply Riesz-Thorin (or Marcinkiewicz) Interpolation when dealing with lower bounds of $\|T\|$ instead of upper bounds? My motivation for answering this question, was to assume that these interpolation theorems can generalize to the reverse inequality. Thus, I was looking for an operator $T$ so that there exists functions $f_k$ with $\|f_k\|_\infty=1$ but $\|Tf_k\|_\infty\to 0$. Here is such a counter-example
Let $X=[0,\infty)$ with the Lebesgue measure. For $x\in [0,\infty)$ write $m$ to be the largest integer such that
$$
n_m:=\sum_{j=0}^m j \leq x.
$$
For a $f\in L^1$ (could make this $L^2$ with a slight modification), let $Tf$ be defined as
$$
Tf(x) = \frac{1}{(m+1)}f\left(\frac{x-n_m}{m+1}+m\right).
$$
Basically, $T$ takes the values of $f$ over the interval $[m,m+1)$ and compresses it by a factor of $m+1$ while stretching it by the same factor. Note that $T$ is linear. Then by a change of variables
\begin{align*}
\|Tf\|_1&=\int_0^\infty |Tf(x)|\,dx=\sum_{m=0}^\infty \int_{n_m}^{n_{m+1}}\frac{1}{(m+1)}\left|f\left(\frac{x-n_m}{m+1}+m\right)\right|\,dx\\&=\sum_{m=0}^\infty\int_{m}^{m+1}|f(y)|\,dy=\int_0^\infty |f(y)|\,dy=\|f\|_1.
\end{align*}
Now for any $k\in \mathbb{N}$, let $f_k=\chi_{[k,k+1)}$ so that $\|f_k\|_p=1$ while $\|f_k\|_p=(k+1)^{1-1/p}\to 0$ as $k\to \infty$. So there is no constant $C_p$ for which
$$
\|f\|_p\leq C_p\|Tf\|_p
$$
for all $f\in L^p$.
