Carleson hunt does not include the lone endpoint

Put $Mf={\sup }_{N\ge 0}\mathrm{\mid }{S}_{N}f\mathrm{\mid }$. Each finite partial sum is measurable, hence this countable supremum is measurable. By F1 it equals infinity outside a null set on a space of measure one. For every positive integer k, the nonnegative simple function k on that conull set is bounded above by Mf, so the definition of the nonnegative integral gives $\int Mf\ge k$. Consequently $\mathrm{\parallel }Mf{\mathrm{\parallel }}_1=\mathrm{\infty }$.

Since $\mathrm{\parallel }f{\mathrm{\parallel }}_1<\mathrm{\infty }$ by F1, step 1.1 contradicts every proposed finite-constant inequality $\mathrm{\parallel }Mh{\mathrm{\parallel }}_1\le C\mathrm{\parallel }h{\mathrm{\parallel }}_1$ for all $h\in L^1$. It also contradicts every finite weak-(1,1) constant: $m\{Mf>t\}=1$ for all t, whereas $C\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{/}t<1$ for sufficiently large t. Thus neither assertion follows by adjoining the endpoint to the proposed positive-p range.