Research proof index

Let $\mathfrak F$ be a forest at level $n$, let $A\subseteq G$ be measurable, and let $f$ be measurable with $\lVert f(x)\rVert\le\mathbf 1_F(x)$. The integral over $A$ of the norm of the total forest Carleson sum is bounded by

$C(a,q,n)\,\mathrm{dens}_2(\mathfrak F)^{1/q-1/2}\,\lVert f\rVert_2\,\mu(A)^{1/2}.$