I am trying to prove a form of the Marcinkiewicz Interpolation Theorem.

$T$ mapping a measurable function to a measurable function is sublinear if $$|T(f_1 + f_2)(x)| \leq |T(f_1)(x)| + |T(f_2)(x)|$$

Attempting to prove the following: Given a sublinear operator $T$ that satisfying

(1)$\mu(\{x: |Tf(x)|> \alpha \}) \leq \frac{C_1}{\alpha} ||f||_{L^1}$

for some finite $C_1$.

(2) $T$ bounded on $L^q$ for some $q >1$: $$||Tf||_{L^q} \leq C_q||f||_{L^q}$$ for some finite $C_q$.

Then $T$ is bounded for all $p$ satisfying $1 < p \leq q$ with $$||Tf||_{L^p} \leq C_p ||f||_{L^p}$$ for some $C_p$ depending only on $p, C_1, C_q$.

I have been given the hint that this can be proved using the first property along with the following: $$||Tf||^p_{L^p} = p \int_0^{\infty} \lambda^{p-1} \mu(\{x: |Tf(x)|> \lambda \})d\lambda $$

Unfortunately, I am unable to prove the theorem though. The $p$ in the above result seems to be the roadblock in the attempts I make. Also, can an estimate on $C_p$ be found?


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