On pointwise a.e. convergence of multilinear operators

TL;DR

This paper proves pointwise almost everywhere convergence for certain multilinear operators, including truncated singular integrals and lacunary multipliers, based on their boundedness properties.

Contribution

It establishes a.e. convergence results for two classes of multilinear operators using maximal operator bounds, advancing understanding of their pointwise behavior.

Findings

Proved a.e. convergence for truncated homogeneous singular integrals.

Established a.e. convergence for lacunary multiplier operators.

Linked convergence to boundedness of associated maximal multilinear operators.

Abstract

In this work we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited decay. The a.e. convergence is deduced from the $L^2\times\cdots\times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.