Sergi Arias: Endpoint estimates for bilinear operators

Abstract

The present thesis is based on the material presented in three research papers, whose main goal is to obtain endpoint estimates for bilinear pseudodifferential operators. In particular, the study is focused on obtaining several estimates involving the endpoint space of functions with local bounded mean oscillation, denoted by \(\mathrm{bmo}(\mathbb{R}^n)\).

In Paper I we establish boundedness properties for bilinear Coifman-Meyer multipliers in the product spaces \(H^1(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)\) and \(L^p(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)\), with \(1<p<\infty\). As a consequence, we are able to study the pointwise product of a function in \(\mathrm{bmo}(\mathbb{R}^n)\) with functions in the Hardy space \(H^1(\mathbb{R}^n)\), in the local Hardy space \(h^1(\mathbb{R}^n)\) and in \(L^p(\mathbb{R}^n)\), with \(1<p<\infty\).

Paper II is devoted to the study of endpoint estimates for bilinear pseudodifferential operators with symbol in the bilinear Hörmander class \(BS^m_{1,1}\), involving \(\mathrm{bmo}(\mathbb{R}^n)\) In combination with the estimates in Paper I, we obtain fractional Leibniz rules for the product of a function in \(\mathrm{bmo}(\mathbb{R}^n)\) and a function in the Hardy space \(h^p(\mathbb{R}^n)\), with \(0<p\leq\infty\).

In Paper III we continue our study on boundedness properties for bilinear pseudodifferential operators with symbol in \(BS^m_{1,1}\). This time, we study the action of those operators on functions in Triebel-Lizorkin spaces of the type \(F^{n/p}_{p,q}(\mathbb{R}^n)\). In particular, we obtain some estimates for the pointwise product of two functions in \(F^{n/p}_{p,q}(\mathbb{R}^n)\) with \(1<p<\infty\), where the spaces involved fail to be multiplicative algebras.