Carleson formalization

adjointCarleson is the adjoint of carlesonOn.

Suppose that, for every error threshold $\delta>0$ and every measure tolerance $\varepsilon>0$, one can choose $N_0$ so that the set where $\sup_{N>N_0}\lVert f(x)-F_N(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$. Then $F_N(x)$ converges to $f(x)$ for almost every $x$.

For an antichain $\mathfrak{A}$ of pairwise incomparable tiles, and measurable functions $f$ and $g$ bounded by the indicators of $F$ and $G$, the pairing of $g$ with the Carleson sum over $\mathfrak{A}$ is controlled by the $L^2$ norms of $f$ and $g$ and by positive powers of the two tile-density parameters. Concretely, the bound is

$C(a,q)\,\mathrm{dens}_1(\mathfrak{A})^{(q-1)/(8a^4)}\,\mathrm{dens}_2(\mathfrak{A})^{1/q-1/2}\,\lVert f\rVert_2\lVert g\rVert_2.$

For an antichain $\mathfrak A$, a measurable set $A\subseteq G$, and measurable $f$ bounded by $\mathbf 1_F$, the norm of the Carleson sum has the integral estimate

$\int_A^+\lVert\operatorname{CarlesonSum}_{\mathfrak A}f(x)\rVert\,dx\le C(a,q)\,\mathrm{dens}_1(\mathfrak A)^{(q-1)/(8a^4)}\,\mathrm{dens}_2(\mathfrak A)^{1/q-1/2}\,\lVert f\rVert_2\,\mu(G)^{1/2}.$

Fix a frequency parameter $\vartheta$, a level $N$, and a spatial grid cube $L$. Among the auxiliary tiles attached to an antichain $\mathfrak{A}$ whose spatial cube is exactly $L$, the total measure of their active sets inside $G$ is at most

$2^{a(N+5)}\,\mathrm{dens}_1(\mathfrak{A})\,\mu(L).$

For a tile $u$, the union of the grid cubes in its level-$n$ boundary family has measure at most a constant $C(X,n)$ times the measure of the spatial cube $\mathcal{I}(u)$.

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.

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

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