Weighted version of Carleson measure and multilinear Fourier multiplier

Abstract

In this paper, we first establish some results about weighted Carleson measure, and then apply them to multilinear Fourier multiplier. By using a different method from the one used by Fujita and Tomita (2012), we obtain some weighted result of multilinear multiplier operators by considering the missing parts of their results in the end-point cases such as from L 2 .!/ L 1 .!/ L 1 .!/ to L 2 .!/, using weighted Carleson measure theory and employing the multilinear interpolation theory.

On the Bilinear Square Fourier Multiplier Operators Associated With Function

“…As an application, they [2] obtained the multiple weighted norm inequality of multilinear Fourier multipliers. For more works about multilinear Fourier multipliers, we refer the reader to [15,20,21]. Recently, Si, Xue and Yabuta [28] considered the bilinear square-function Fourier multiplier operator defined as follows,…”

Endpoint estimates for commutators of mutilinear square function satisfying some integrable condition

“…Later, Grafakos and Torres [9,10] extended Coifman and Meyer's works. Since multilinear Littlewood-Paley g-function and related multilinear Littlewood-Paley type estimates were applied to PDE and any other fields, see [4][5][6], more and more people devote themselves to study the multilinear Littlewood-Paley theory and more works about multilinear Littlewood-Paley type operators, see [13,15,17,18] and the references therein.…”

A counterexample to weighted estimates for multilinear Fourier multipliers with Sobolev regularity

“…where the implicit constant is independent of m (see Li, Xue and Yabuta [14] for the endpoint cases). This result can also be obtained from another approach of Hu and Lin [8].…”