Dens1 le dens

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

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