Carleson hunt maximal inequality on the torus

Choose $r=(p+1)\mathrm{/}2$ and $s=2p$, so $1<r<p<s<\mathrm{\infty }$. Apply F1 at $r$ and $s$, then F2. The resulting finite-model strong $(p,p)$ constant $B_p$ depends only on $p$ and the fixed packet, not on the finite family or measurable selector.

F3 transfers this uniform bound to $\mathrm{\parallel }{C}_{\mathbb{R}}u{\mathrm{\parallel }}_p\le D_p\mathrm{\parallel }u{\mathrm{\parallel }}_p$ for every complex Schwartz $u$, with $D_p={\kappa }^{-1}{B}_p<\mathrm{\infty }$ and the positive reconstruction constant defined there. No passage from weak type at the same exponent is used. Since $p$ was arbitrary, these real-line bounds hold separately for every exponent strictly between one and infinity.

Apply F4 at the chosen $p$. It gives a finite constant $C_p$, depending only on $p$ and the real-line bound $D_p$, such that $\mathrm{\parallel }{\sup }_{N\ge 0}\mathrm{\mid }{S}_{N}f\mathrm{\mid }{\mathrm{\parallel }}_{L^p(\mathbb{T})}\le C_p\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathbb{T})}$ with precisely the coefficients and normalized measure in F5. The countable supremum is measurable, and the estimate holds for the actual partial sums of every complex $L^p$ representative class; changing the representative does not change any coefficient. The case $f=0$ and the cutoff $N=0$ are included by F4. The argument uses no $p=1$ or $p=\mathrm{\infty }$ assertion. AC is inherited exactly through F6 and its suppliers.