Marcinkiewicz interpolation from weak $(1,1)$ and strong $(\mathrm{\infty },\mathrm{\infty })$

Fix $t>0$ and $\eta >0$, and put $C:=B+\eta >0$. Since the strong [L1, given, construct, algebra] $(\mathrm{\infty },\mathrm{\infty })$ bound with constant $B$ also holds with the larger constant $C$, split $f=f_t^{>}+f_t^{\le },f_t^{>}:=f{\mathbf{1}}_{\{\mathrm{\mid }f\mathrm{\mid }>t\mathrm{/}(2C)\}},f_t^{\le }:=f{\mathbf{1}}_{\{\mathrm{\mid }f\mathrm{\mid }\le t\mathrm{/}(2C)\}}\mathrm{.}$ Sublinearity gives $\mathrm{\mid }Tf\mathrm{\mid }\le \mathrm{\mid }T{f}_t^{>}\mathrm{\mid }+\mathrm{\mid }T{f}_t^{\le }\mathrm{\mid }\mathrm{.}$ Since $\mathrm{\parallel }{f}_t^{\le }{\mathrm{\parallel }}_{\mathrm{\infty }}\le t\mathrm{/}(2C)$, the strong $(\mathrm{\infty },\mathrm{\infty })$ bound with constant $C$ yields $\mathrm{\mid }T{f}_t^{\le }\mathrm{\mid }\le C\mathrm{\parallel }{f}_t^{\le }{\mathrm{\parallel }}_{\mathrm{\infty }}\le t\mathrm{/}2.$ Therefore $\{\mathrm{\mid }Tf\mathrm{\mid }>t\}\subseteq \{\mathrm{\mid }T{f}_t^{>}\mathrm{\mid }>t\mathrm{/}2\}\mathrm{.}$

On $\{\mathrm{\mid }f\mathrm{\mid }>t\mathrm{/}(2C)\}$ one has $\mathrm{\mid }f\mathrm{\mid }\le (2C\mathrm{/}t{)}^{p-1}\mathrm{\mid }f{\mathrm{\mid }}^p$, so $f_t^{>}\in L^1$ because $f\in L^p$. Apply the weak $(1,1)$ bound: [L1, step 1.1, algebra] $\mu (\{\mathrm{\mid }Tf\mathrm{\mid }>t\})\le \mu (\{\mathrm{\mid }T{f}_t^{>}\mathrm{\mid }>t\mathrm{/}2\})\le \frac{2A}{t}\mathrm{\parallel }{f}_t^{>}{\mathrm{\parallel }}_1=\frac{2A}{t}{\int }_{\{\mathrm{\mid }f\mathrm{\mid }>t\mathrm{/}(2C)\}}\mathrm{\mid }f\mathrm{\mid }d\mu \mathrm{.}$

Using [L3] with $g=Tf$ and then step 2.1, [L2, L3, step 2.1, algebra] $\mathrm{\parallel }Tf{\mathrm{\parallel }}_p^p=p{\int }_0^{\mathrm{\infty }}{t}^{p-1}\mu (\{\mathrm{\mid }Tf\mathrm{\mid }>t\})dt\le 2Ap{\int }_0^{\mathrm{\infty }}{t}^{p-2}\left({\int }_{\{\mathrm{\mid }f\mathrm{\mid }>t\mathrm{/}(2C)\}}\mathrm{\mid }f\mathrm{\mid }d\mu \right)dt\mathrm{.}$

Define the positive measure $\lambda (E)={\int }_E\mathrm{\mid }f\mathrm{\mid }d\mu$ by [L4]. Apply [L3] with exponent $p-1>0$, function $\mathrm{\mid }f\mathrm{\mid }$, and measure $\lambda$; then substitute $t=2Cu$ in the one-dimensional nonnegative integral from step 3.1. This gives ${\int }_0^{\mathrm{\infty }}{t}^{p-2}\lambda (\{\mathrm{\mid }f\mathrm{\mid }>t\mathrm{/}(2C)\})dt=\frac{(2C{)}^{p-1}}{p-1}{\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p-1}d\lambda \mathrm{.}$ By [L4], the last integral is ${\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu$. Hence $\mathrm{\parallel }Tf{\mathrm{\parallel }}_p^p\le \frac{2^{p}Ap}{p-1}{C}^{p-1}{\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu \mathrm{.}$ This uses no sigma-finiteness assumption on $\mu$.

Taking $p$th roots in step 4.1 gives [step 4.1, algebra] $\mathrm{\parallel }Tf{\mathrm{\parallel }}_p\le 2{\left(\frac{Ap}{p-1}\right)}^{1\mathrm{/}p}{C}^{1-1\mathrm{/}p}\mathrm{\parallel }f{\mathrm{\parallel }}_p\mathrm{.}$ Because $\eta >0$ was arbitrary, letting $\eta \downarrow 0$ yields $\mathrm{\parallel }Tf{\mathrm{\parallel }}_p\le 2{\left(\frac{Ap}{p-1}\right)}^{1\mathrm{/}p}{B}^{1-1\mathrm{/}p}\mathrm{\parallel }f{\mathrm{\parallel }}_p\mathrm{.}$ Thus $T$ is of strong type $(p,p)$ for every $1<p<\mathrm{\infty }$.