What the Carleson–Hunt proof requires

Assume the Axiom of Choice (The Axiom of Choice) for the Fourier and measure interfaces of the following proved suppliers. The local conclusion is Carleson hunt maximal inequality on the torus, a strong maximal estimate for symmetric partial sums on the normalized period-one torus, at each $1<p<\mathrm{\infty }$.

The real-line tile argument uses Carleson density selection and Carleson size selection to select forests and Carleson single tree estimate to bound a tree contribution. The exceptional-set argument and distribution bootstrap of Hunt exceptional set and distribution estimates give finite tile models uniform restricted weak type at every exponent strictly between one and infinity. Then Carleson restricted weak interpolation gives uniform strong bounds at the desired exponent.

The packet averaging and reconstruction argument in Wave packet model dominates the linearised carleson operator passes those model bounds to the real-line Carleson maximal operator on Schwartz inputs. Finally Carleson real line to torus transfer uses a slowly varying Schwartz window on trigonometric polynomials, periodic averaging, Fejer approximation and monotone convergence to obtain the torus estimate. This supplies the transference argument as well as the time-frequency model estimates. The local theorem assembles these interfaces; the assumption AC is inherited from them.