Endpoint estimate for Calderón–Zygmund operators in the multiparameter flag setting

We establish the weak endpoint estimate: for every λ > 0 , | { ( x , y ) ∈ R n × R m : | T f ( x , y ) | > λ } | ≤ C ∬ R n + m | f ( x , y ) | λ ( 1 + ( log + ⁡ | f ( x , y ) | λ ) ) d x d y , where T denotes the area integral, square function, and the vertical and non-tangential maximal operators, and the multiparameter flag singular integrals in the setting of Nagel–Ricci–Stein. The ideas of the proofs stem from C. Fefferman and Stein's celebrated good- λ inequality and R. Fefferman's decomposition of L log + ⁡ L ( R n × R m ) functions (supported in the unit cube) into bump functions; our approach develops flag analogues of these tools in the Nagel–Ricci–Stein setting.