Marcinkiewicz Theorem says that, if $1\le p<q$ given a bounded operator $T$ from $L^p(\Bbb R^n)$ to $L^{p,w}(\Bbb R^n)$ (the last one is the $L^p$ weak space) AND EVEN from $L^q(\Bbb R^n)$ to $L^{q,w}(\Bbb R^n)$, then $T$ is also a bounded operator from $L^r$ into inself, for every $r\in]p,q[$.
My problem is: all these spaces don't form a chain, there is no hierarchy of inclusion between them (since they are considered on the whole $\Bbb R^n$), thus I can't conclude anything like $L^r(\Bbb R^n)\subseteq L^p(\Bbb R^n)\cap L^q(\Bbb R^n)$. Thus, why is $T$ automatically defined even on the $L^r(\Bbb R^n)$ spaces?
Many thanks!
We have natural inclusions $L^p \cap L^q \subset L^r \subset L^p + L^q$, if $1 \leq p < r < q \leq \infty$. Since $T$ is defined on both $L^p$ and $L^q$, it is defined on their intersection and sum, and hence it is defined on $L^r$.