$L^2$ estimates for the operators $ \bar\partial $ and $ \bar\partial_b $

• Ano de defesa: 2019
• Autor(a) principal: Coacalle, Joel Rogelio Portada
• Orientador(a): Hoepfner, Gustavo
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de São Carlos; Câmpus São Carlos
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática - PPGM
• Palavras-chave em Português: Operador de Cauchy Riemann; Operador Tangencial de Cauchy Riemann; Variedade CR; Condição Y(q) fraca; Estimativas de imagem fechada
• Palavras-chave em Inglês: Cauchy Riemann operator; Tangential Cauchy Riemann operator; CR manifolds; Weak Y(q) condition; Closed range estimates
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: https://repositorio.ufscar.br/handle/ufscar/12022

The purpose of this work is to establish sufficient conditions for closed range estimates on $(0,q)$-forms, for some fixed $q$, $1 \leq q \leq n-1$, for $\bar\partial_b$ in both $L^2$ and $L^2$-Sobolev spaces in embedded, not necessarily pseudoconvex CR manifolds of hypersurface type. The condition, named weak $Y(q)$, is both more general than previously established sufficient conditions and easier to check. Applications of our estimates include estimates for the Szeg\"o projection as well as an argument that the harmonic forms have the same regularity as the complex Green operator. We use a microlocal argument and carefully construct a norm that is well-suited for a microlocal decomposition of form. We do not require that the CR manifold is the boundary of a domain. Finally, we provide an example that demonstrates that weak $Y(q)$ is an easier condition to verify than earlier, less general conditions.