Perturbações de operadores globalmente Gevrey hipoelíticos

Abstract

Resumo: Neste trabalho estudamos a influˆencia de termos de ordem zero (perturba?c˜oes) sobre a hipoeliticidade Gs global de operadores da forma ?t ? c(t)?x, supondo que c(t) 'e uma fun?c˜ao Gevrey de vari'avel real. Mostramos que quando Im(c(t)) n˜ao muda de sinal e n˜ao 'e identicamente nula, o operador perturbado continua sendo globalmente Gs hipoel '?tico; e quando Im(c(t)) 'e identicamente nula, a hipoeliticidade Gs global do operador perturbado 'e equivalente a hipoeliticidade Gs global de um operador a coeficientes constantes conjugado. Tamb'em analisamos perturba?c˜oes em espa?cos de dimens˜ao maior que 2 obtendo uma interessante rela?c˜ao entre a hipoeliticidade Gs global de do operador perturbado e a injetividade de um operador conjugado. Palavras-chave: Fun?c˜oes Gevrey Peri'odicas, Hipoeliticidade, Resolubilidade, Perturba ?c˜oes, N'umeros de Liouville .

Abstract: In this work we study the influence of zero order terms (perturbations) on the global Gs hypoelipticity of operators in the form ?t ? c(t)?x, assuming that c(t) is a Gevrey function of real variable. We show that when Im(c(t)) does not change sign and is not identically null, the perturbed operator remains globally Gs hypoeliptic, and when Im(c(t)) is null, the global Gs hypoelipticity of the perturbed operator is equivalent to the global Gs hypoelipticity of a constant coefficient operator conjugated. We also analyze perturbations in spaces of dimension greater than two obtaining an interesting relation between the global Gs hypoelipticity of the perturbed operator and the injectivity of a conjugated operator. Keywords: Periodic Gevrey Functions, Hypoelipticity, Solvability, Perturbations, Liouville Numbers.