We present a new proof of the classical weak-type (1, 1) estimate for Calderón-Zygmund operators. This proof is inspired by ideas of Nazarov, Treil, and Volberg that address the non-doubling setting. An application to a weighted weak-type inequality is also given.

Two proofs of a weighted weak-type 1, . . . , 1; 1 m estimate for multilinear Calderón-Zygmund operators are given. The ideas are motivated by different proofs of the classical weak-type (1, 1) estimate for Calderón-Zygmund operators. One proof uses the Calderón-Zygmund decomposition, and the other proof is motivated by ideas of Nazarov, Treil, and Volberg.

Starting from the well-established notion of a separating family (or separating system) and the refinement known as a splitting family, we define and study generalizations called $n$-separating and $n$-splitting families, obtaining lower and upper bounds on their minimum sizes. For $n$-separating families our bounds are asymptotically tight within a linear factor, while for $n$-splitting families we provide partial results and open questions.

By means of appropriate sparse bounds, we deduce compactness on weighted L p (w) spaces, 1 < p < ∞, for all Calderón-Zygmund operators having compact extensions on L 2 (R n ). Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces.

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