Research proof index

The real-line Carleson operator is positively homogeneous. For every $a>0$,

$T f(x)=a\,T(f/a)(x),$

where the scalar on the right is interpreted in the extended nonnegative reals.

For every continuous, $2\pi$-periodic function $f : \mathbb{R} \to \mathbb{C}$, the symmetric partial Fourier sums $S_N f(x)$ converge to $f(x)$ for almost every $x \in \mathbb{R}$.

Let $T>0$, $1 \le p < \infty$, and let $f$ belong to $L^p((0,T])$. For every $\varepsilon>0$, there is a smooth $T$-periodic function $f_0 : \mathbb{R}\to\mathbb{C}$ such that

$\lVert f-f_0\rVert_{L^p((0,T])} \le \varepsilon.$

In a conditionally complete linear order, suppose the values $f(i)$ for $i\in s$ are bounded above. If one of those values is exactly $a$, then $a$ is at most the supremum $\sup_{i\in s} f(i)$.

For a fixed spatial grid cube $I$, every frequency parameter lying in one of the prescribed balls centered at the finite net $\mathcal Z(I)$ also lies in at least one of the frequency regions $\Omega_1(I,f)$. Equivalently, the union of those net balls is contained in the union of the first-stage tile frequency regions.

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 $\varepsilon$.