Research proof index

There is a measurable exceptional set $G'\subseteq G$ with $2\mu(G')\le\mu(G)$ such that every measurable $f$ bounded by $\mathbf{1}_F$ satisfies

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

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. Assume $a\ge4$ and that the truncated Calderón-Zygmund operators $T_r$ satisfy the required uniform strong $L^2$ estimate for every $r>0$. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$, then the two-sided metric Carleson operator satisfies

$\int_G^+ \mathcal C_K f(x)\,dx\le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$