Sobolev spaces and hyperbolic fillings

الأصل · EN

Let Z be an Ahlfors Q-regular compact metric measure space, where Q>0. For p>1 we introduce a new (fractional) Sobolev space Aᵖ(Z) consisting of functions whose extensions to the hyperbolic filling of Z satisfies a weak-type gradient condition. If Z supports a Q-Poincaré inequality with Q>1, then AQ(Z) coincides with the familiar (homogeneous) Hajłasz-Sobolev space.