Caracterização da continuidade de operatores do tipo Calderón-Zygmund fortemente singular em espaços de Hardy

Abstract

In this thesis, we characterize the continuity of strongly singular Calderón-Zygmund operators of type $\sigma$ in Hardy spaces, weighted Hardy spaces and Hardy-Morrey spaces in the spirit of (Coifman and Meyer, 1977). In particular, we consider weaker integral Hormander-type conditions on the kernel. Calderón-Zygmund operators of this type include appropriate classes of pseudodifferential operators $OpS^{m}_{\sigma,\nu}(\Rn)$ and operators associated to standard $\delta$-kernels of type $\sigma$ introduced by Álvarez and Milman in (Álvarez and Milman, 1986).

The method to obtain the boundedness properties refers to the atomic and molecular decomposition of such spaces. In particular, in order to obtain it for local Hardy spaces $h^p(\R^n)$ for $0<p\leq 1$, we present a new approach to atoms and molecules assuming weaker cancellation conditions, extending and unifying previous results presented in (Dafni, 1993), (Komori, 2001), (Dafni and Yue, 2012) and (Dafni and Liflyand, 2019). As applications, we prove a non-homogeneous version of Hardy's inequality in $h^p(\Rn)$ and improved necessary and sufficient conditions for the continuity of inhomogeneous Calderón-Zygmund type operators on these spaces.