Research proof index

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

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