Research proof index

Fix $1<p<2$, an error threshold $\delta>0$, and a measure tolerance $\varepsilon\ge0$. There is an explicit bound $C(\delta,\varepsilon,p)$ such that, whenever a measurable $2\pi$-periodic function $g$ satisfies $\lVert g\rVert_{L^p((0,2\pi])}\le C(\delta,\varepsilon,p)$, the set where the maximal partial Fourier sum $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.