Стаття є оглядом сучасних результатiв, присвячених теорiї елiптичних операторiв i елiптичних крайових задач у гiльбертових шкалах, що складаються з функцiональних просторiв Хермандера. Встановлено теореми про нетеровiсть та локальну регулярнiсть розв’язкiв. Наведено застосування до питань збiжностi спектральних розвинень за власними функцiями елiптичних операторiв.

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