Endpoint targets: weak (1,1) here, L∞ to BMO later; strong endpoints fail in general

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). For an $L^2$-bounded Calderón–Zygmund operator the two endpoint-adjacent facts established on this page are the weak $(1,1)$ estimate of Calderón–Zygmund operators are of weak type (1,1) and the strong $L^p$ bounds for $1<p<\mathrm{\infty }$ of Calderón–Zygmund operators are bounded on Lp. The following claims are deliberately not made here, and the reader should not read the page as asserting or refuting them.

First, no strong type $(1,1)$ bound and no bounded action on $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ is claimed; the weak $(1,1)$ estimate is the endpoint substitute for the former, and the companion examples page exhibits the Hilbert transform of an interval indicator as a counterexample to compatible strong $L^1$ and $L^{\mathrm{\infty }}$ conclusions, for the Hilbert transform and hence for the class of Calderón–Zygmund operators.

Second, the $L^{\mathrm{\infty }}\to \mathrm{B}\mathrm{M}\mathrm{O}$ endpoint is the subject of the later BMO page of this track, where bounded mean oscillation is defined and the endpoint estimate is proved; neither the statement nor the proof of that estimate is used here, and no $L^{\mathrm{\infty }}$ conclusion is available from it.

The strict-range bounds are stated with the exponent range $1<p<\mathrm{\infty }$ only; the constants blow up as $p\downarrow 1$ and as $p\to \mathrm{\infty }$ in the estimates recorded above, consistently with the two refuted endpoints.