Marcinkiewicz interpolation from weak (1,1) and strong (2,2)

Fix a representative of $f$ and, for each height $t>0$, split $f=f_1^{(t)}+f_2^{(t)}$ with $f_1^{(t)}:=f{\mathbf{1}}_{\{\mathrm{\mid }f\mathrm{\mid }>t\}}$ and $f_2^{(t)}:=f{\mathbf{1}}_{\{\mathrm{\mid }f\mathrm{\mid }\le t\}}$; both are measurable, $\mathrm{\parallel }{f}_1^{(t)}{\mathrm{\parallel }}_1={\int }_{\{\mathrm{\mid }f\mathrm{\mid }>t\}}\mathrm{\mid }f\mathrm{\mid }d\mu \le t^{1-p}\mathrm{\parallel }f{\mathrm{\parallel }}_p^p<\mathrm{\infty }$ and $\mathrm{\parallel }{f}_2^{(t)}{\mathrm{\parallel }}_2^2={\int }_{\{\mathrm{\mid }f\mathrm{\mid }\le t\}}\mathrm{\mid }f{\mathrm{\mid }}^{2}d\mu \le t^{2-p}\mathrm{\parallel }f{\mathrm{\parallel }}_p^p<\mathrm{\infty }$, so $f_1^{(t)}\in L^1$, $f_2^{(t)}\in L^2$, and both lie in $L^1+L^2$; moreover $\mathrm{\mid }{f}_2^{(t)}\mathrm{\mid }\le t$ and $f=f_1^{(t)}+f_2^{(t)}$ pointwise.

Sublinearity gives $\mathrm{\mid }Tf\mathrm{\mid }\le \mathrm{\mid }T{f}_1^{(t)}\mathrm{\mid }+\mathrm{\mid }T{f}_2^{(t)}\mathrm{\mid }$ pointwise, hence $\{\mathrm{\mid }Tf\mathrm{\mid }>t\}\subseteq \{\mathrm{\mid }T{f}_1^{(t)}\mathrm{\mid }>t\mathrm{/}2\}\cup \{\mathrm{\mid }T{f}_2^{(t)}\mathrm{\mid }>t\mathrm{/}2\}$, and subadditivity of $\mu$ together with [F1] applied to $f_1^{(t)}$ (weak $(1,1)$ at level $t\mathrm{/}2$) and to $f_2^{(t)}$ (strong $(2,2)$, then [F2] applied to $\mathrm{\mid }T{f}_2^{(t)}{\mathrm{\mid }}^2$ at level $(t\mathrm{/}2{)}^2$) yields $A_{Tf}(t)\le \frac{2A}{t}\parallel f_1^{(t)}{\parallel }_1+\frac{4B^2}{t^2}\parallel f_2^{(t)}{\parallel }_2^2=\frac{2A}{t}{\int }_{\{\mathrm{\mid }f\mathrm{\mid }>t\}}\mathrm{\mid }f\mathrm{\mid }d\mu +\frac{4B^2}{t^2}{\int }_{\{\mathrm{\mid }f\mathrm{\mid }\le t\}}\mathrm{\mid }f{\mathrm{\mid }}^{2}d\mu \mathrm{.}$

Tonelli's theorem applied to the nonnegative product-measurable function $(x,t)\mapsto t^{p-2}\mathrm{\mid }f(x)\mathrm{\mid }{\mathbf{1}}_{\{\mathrm{\mid }f(x)\mathrm{\mid }>t\}}$ on the $\sigma$-finite product $X\times (0,\mathrm{\infty })$ converts the first term of the layer-cake integral into an $X$-integral: ${\int }_0^{\mathrm{\infty }}{t}^{p-1}\frac{2A}{t}\left({\int }_{\{\mathrm{\mid }f\mathrm{\mid }>t\}}\mathrm{\mid }f\mathrm{\mid }d\mu \right)dt=2A{\int }_X\mathrm{\mid }f\mathrm{\mid }\left({\int }_0^{\mathrm{\mid }f\mathrm{\mid }}{t}^{p-2}dt\right)d\mu =\frac{2A}{p-1}{\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu ,$ where the inner integral was evaluated as $\mathrm{\mid }f{\mathrm{\mid }}^{p-1}\mathrm{/}(p-1)$, legitimate because $p-1>0$, and the case $f(x)=0$ contributes $0$.

Likewise, Tonelli applied to $(x,t)\mapsto t^{p-3}\mathrm{\mid }f(x){\mathrm{\mid }}^2{\mathbf{1}}_{\{\mathrm{\mid }f(x)\mathrm{\mid }\le t\}}$ gives, since $p-2<0$, ${\int }_0^{\mathrm{\infty }}{t}^{p-1}\frac{4B^2}{t^2}\left({\int }_{\{\mathrm{\mid }f\mathrm{\mid }\le t\}}\mathrm{\mid }f{\mathrm{\mid }}^{2}d\mu \right)dt=4B^2{\int }_X\mathrm{\mid }f{\mathrm{\mid }}^2\left({\int }_{\mathrm{\mid }f\mathrm{\mid }}^{\mathrm{\infty }}{t}^{p-3}dt\right)d\mu =\frac{4B^2}{2-p}{\int }_X\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu ,$ the inner integral being $\mathrm{\mid }f{\mathrm{\mid }}^{p-2}\mathrm{/}(2-p)$ and the case $f(x)=0$ contributing $0$.

Combining the layer-cake identity [F3] for $g=Tf$ with step 2.1 and steps 3.1 and 3.2 gives $\mathrm{\parallel }Tf{\mathrm{\parallel }}_p^p=p{\int }_0^{\mathrm{\infty }}{t}^{p-1}{A}_{Tf}(t)dt\le p(\frac{2A}{p-1}+\frac{4B^2}{2-p})\mathrm{\parallel }f{\mathrm{\parallel }}_p^p,$ because $t^{p-1}{A}_{Tf}(t)\le t^{p-1}$ times the two integrands integrated in steps 3.1 and 3.2; taking $p$-th roots gives the asserted strong $(p,p)$ bound. Since $Tf$ is determined a.e. by the class of $f$, the bound descends to $L^p(X)$ by [F5].