Carleson hunt maximal inequality on the torus

Choose $r=(p+1)\mathrm{/}2$ and s=2p, so1<r<p<s<infinity. 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 Lp 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=infinity assertion. AC is inherited exactly through F6 and its suppliers.