New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory

Abstract

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.

Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators

“…Moreover, the regularity assumptions are also good enough to obtain a weighted theory for classical A p weights, as shown by Grafakos-Torres [12] and Pérez-Torres [22], which was later extended to new optimal multilinear classes of weights by Lerner-Ombrosi-Pérez-Torres-Trujillo [18].…”

Commutators of multilinear Calderón–Zygmund operators with Dini type kernels on some function spaces

“…Obviously, when ω(t) = t ε for some ε > 0, the m-linear ω-CZO is exactly the multilinear Calderón-Zygmund operator studied by Grafakos and Torres [6] and Lerner et al [9].…”

Calderón‐Zygmund operators with non‐diagonal singularity

“…And it has been widely studied in various directions, e.g., see [1-5, 8, 9, 12] and references therein. We refer to [14][15][16][17][19][20][21][22][23] for some recent advances of the Calderón-Zygmund theory.…”