Research proof index

For $x\in[0,2\pi]$ and an interval-integrable function $g$, the norm of the localized Dirichlet-kernel integral over $[x-\pi,x+\pi]$, with cutoff $\max(1-|x-y|,0)$, is bounded by the sum of the real Carleson operators applied to $g$ and to its complex conjugate.