2021/11/11 by Michel Alexis, Eric T. Sawyer, Alexis, Michel +3
Mathematics · #Advanced Harmonic Analysis Research #Holomorphic and Operator Theory #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2111.06277
We consider the weak to strong type problem for two weight norm inequalities for Calderón-Zygmund operators with doubling weights. We show that if a Calderón-Zygmund operator T is weak type (2,2) with doubling weights, then it is strong type (2,2) if and only if the dual cube testing condition for T* holds, alternatively if and only if the dual cancellation condition of Stein holds. The testing condition can be taken with respect to either cubes or balls, and more generally, this is extended to a weak form of Tb theorem. Finally, we show that for all pairs of locally finite positive Borel measures, and all Stein elliptic Calderón-Zygmund operators T, the weak type (2,2) inequalities for T and and its associated maximal truncations operator T* are equivalent. Thus the characterization of weak type for T* in [LaSaUr1] applies to T as well.