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A typical way to show $L^{p}$ boundedness of a linear or sub-linear operator is to show a weak type $(1,1)$ bound and an $L^{p}$ bound for some $p$, and then combine these two using interpolation to prove $L^{p}$ bounds for all $p$ between $1$ and $\infty$. A typical example is that of the Hardy-Littlewood maximal Operator. Are there any examples\techniques where $L^{p}$ boundedness is proved without establishing some type of weak type bounds?

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One can work directly with $L^p$ (not using interpolation), and there is a good reason to do so when a sharp bound for the operator norm is desired. See, for example, The Hunt for a Bellman function by Nazarov and Treil.

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