We will discuss the connection between the study of non-local operators on Eu-

clidean space to the study of a class of fractional GJMS operators in conformal

geometry. The emphasis is on the study of a class of fourth order operators and

their third order boundary operators. We will also discuss geometry tools (e.g. met-

ric with measures), the positivity property and Sobolev trace inequalities associated

with these operators via extension theorems. This is a report of joint works with

Jeffrey Case, and separately with Antonio Ache.