Research proof index

Let $\mathfrak A$ be an antichain contained in the tile class $\mathfrak C(k,n)$, and let $f$ be measurable with $\lVert f(x)\rVert\le\mathbf 1_F(x)$. On $G\setminus G'$, the Carleson sum over the selected positive tiles in $\mathfrak A$ has an $L^1$ bound with exponential decay in the layer index $n$:

$\int_{G\setminus G'}^+\!\lVert\operatorname{CarlesonSum}f(x)\rVert\,dx\le C(a,q)\,\mu(G)^{1-1/q}\mu(F)^{1/q}\,2^{-((q-1)/(8a^4))n}.$