Interpolation theorems I am trying to understand a result from the Marcinkiewics interpolation theorem. It goes as follows.
If $T$ is sublinear (1,1)-weak type and bounded in $L^\infty(R^n)$ with norm 1. I am trying to show that
there is $C>0$ so that
$$ \int_B |Tf|dx\le Cm(B(0,1))+C\int_{R^n} |f|\log^+|f|dm$$
where $m$ is the Lebesgue measure. I am looking for a reference for a proof for this result as I do not see how to apply either interpolation theorem.
 A: Let $B:=B(0, 1)$ (only the boundedness of the set is needed I guess). Write $f=f_1+f_2$ where $f_1(x)=f(x)$ if $|f(x)|>t$ and $f_1(x)=0$ otherwise. Then by the sublinearity of $T$
\begin{align*}
|T f|&\leq |T f_1|+ |T f_2|\\
     &\leq |Tf_1|+t,
\end{align*}
since $|f_2|\leq t$ implies that $|Tf_2|\leq t$ ($T$ is bounded in $L^\infty$). From the last estimate it follows the set inclusion $\{|T f|>2t\}\subset \{|T f_1|>t\}$, which together with the fact that $T$ is of weak type $(1, 1)$ implies
\begin{align*}
\mu(\{|Tf|>2t\})&\leq \mu(\{|T f_1|>t\})\\
&\leq \dfrac{C}{t}\int_{\mathbb{R}^n} |f_1(x)|dx\\
&= \dfrac{C}{t}\int_{\{|f|>t\}} |f(x)|dx.
\end{align*}
Then by using the Layer-cake representation and Fubini's theorem
\begin{align*}
\int_B |T f(x)|dx&=\int_0^\infty m(\{x\in B: |T f(x)|>t\})dt\\
&=2\int_0^\infty m(\{x\in B: |T f(x)|>2t\})dt\\
&\leq 2\int_0^1 m(B)dt+2\int_1^\infty \mu(\{|T f|>2t\})dt\\
&\leq 2m(B)+\int_1^\infty \dfrac{C}{t}\left(\int_{\{|f|>t/2\}} |f(x)|dx\right)dt\\
&=2m(B)+C\int_{\mathbb{R}^n} |f(x)|\left(\int_1^{|f(x)|} \dfrac{1}{t}dt\right) dx\\
&=2m(B)+C\int_{\mathbb{R}^n} |f(x)|\log^+|f(x)|dx,
\end{align*}
where $\log^+(t)=\max\{\log t, 0\}$.
EDIT: I don't know how to get the result as an application of Marcinkiewicz nor a reference of it, but at least I hope that this answer help you anyway.
