Interpolate L1 to Linfinity and L2 to L2 bounds

For $1<p<2$, set $\theta =2-2\mathrm{/}p$. Then $0<\theta <1$, $(1-\theta )\mathrm{/}1+\theta \mathrm{/}2=1\mathrm{/}p$, and $(1-\theta )\mathrm{/}\mathrm{\infty }+\theta \mathrm{/}2=1-1\mathrm{/}p=1\mathrm{/}{p}^{\mathrm{\prime }}$. Thus the core interpolation theorem with endpoints $(1,\mathrm{\infty })$ and $(2,2)$ gives exactly $A^{1-\theta }{B}^{\theta }=A^{2\mathrm{/}p-1}{B}^{2-2\mathrm{/}p}$ and proves membership in $L^{p^{\mathrm{\prime }}}$. Zero A or B is allowed because the interior powers are positive.

At p=1 the target is infinity and at p=2 the target is two, so the asserted estimates are the respective hypotheses. Assuming countable choice, the compatible-extension result applies with finite source endpoint exponents 1 and 2 and the given sigma-finite spaces, providing the claimed unique bounded extensions and agreement.