A note on weighted bounds for singular operators with nonsmooth kernels

The Anh Bui¹ ; José M. Conde-Alonso² ; Xuan Thinh Duong¹ ; Mahdi Hormozi³

¹ Department of Mathematics Macquarie University North Ryde, NSW 2109, Australia
² Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones Científicas C/ Nicolás Cabrera 13-15 28049 Madrid, Spain
³ Department of Mathematical Sciences Division of Mathematics University of Gothenburg 41296 Gothenburg, Sweden and Department of Mathematics Shiraz University Shiraz 71454, Iran

Studia Mathematica, Tome 236 (2017) no. 3, pp. 245-269

Résumé

Let $T$ be a multilinear {integral} operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual Hölder continuity of kernels of multilinear Calderón–Zygmund singular integral operators. In this paper, given a suitable multiple weight $\vec{w}$, we obtain a bound for the weighted norm of $T$ in terms of $\vec{w}$. As applications, we obtain new weighted bounds for certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schrödinger operators on $\mathbb {R}^n$.

Détail

DOI : 10.4064/sm8409-9-2016

Keywords: multilinear integral operator which bounded certain products lebesgue spaces mathbb assume its associated kernel satisfies mild regularity condition which weaker usual lder continuity kernels multilinear calder zygmund singular integral operators paper given suitable multiple weight vec obtain bound weighted norm terms vec applications obtain weighted bounds certain singular integral operators linear multilinear fourier multipliers riesz transforms associated schr dinger operators nbsp mathbb