Calderón–Zygmund decompositions in tent spaces and weak-type endpoint bounds for two quadratic functionals of Stein and Fefferman–Stein

On regularity of weak solutions to linear parabolic systems with measurable coefficients

“…For p < 2, we use another approach, based on the following Calderón-Zygmund decomposition of tent space functions, discovered by Huang in his PhD thesis (and published in [35]). Lemma 4.3.…”

“…Extrapolation for p < 2, however, requires another argument than for the Lions operator as we do not know how to reach p = 1, and hence cannot use atomic decompositions. We use a Calderón-Zygmund decomposition in tent spaces introduced by Huang in his PhD thesis, see [35], and prove weak type T p estimates. Such a decomposition is natural, but difficult to use, because of the lack of cancellation of local terms in this decomposition.…”