Research proof index

Let $1<q\le2$, let $q'$ be its Hölder conjugate, and suppose the linearized nontangential operators have the required uniform $L^2$ bound. For bounded measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf 1_F(x)$, the supremum over all truncated scale intervals $-B\le s_1\le s_2\le B$ satisfies

$\int_G^+\sup_{s_1\le s_2}\lVert T_{s_1,s_2}f(x)\rVert\,dx\le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$