Abstract Hardy–Sobolev spaces and interpolation

Abstract

Hardy-Sobolev spaces and interpolation AbstractThe purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities.

Real interpolation for grand Besov and Triebel-Lizorkin spaces on RD-spaces

“…It was also pointed out in [10] that KS Some other recent developments on the real interpolation theory of Sobolev spaces on metric spaces were made by Badr [1,2] and Badr-Bernicot [3]. Badr in [1,2] obtained the interpolation properties between two Sobolev spaces both with order 1 on some classes of manifolds, Lie groups and metric spaces satisfying certain doubling properties, while Badr and Bernicot [3] studied the real interpolation between Hardy-Sobolev spaces and Sobolev spaces both with order 1 on doubling Riemannian manifolds via an atomic decomposition. Notice that the Triebel-Lizorkin spaces coincide with Sobolev spaces for parameters s = 1 and certain p, q.…”

An atomic decomposition of the Hajłasz Sobolev spaceM11on manifolds

“…In [5], the authors defined atomic Hardy-Sobolev spaces. Let us recall their definition of homogeneous Hardy-Sobolev atoms.…”

“…Let us recall their definition. [5].) For 1 < t ∞, we say that a function a is a nonhomogeneous Hardy- Sobolev (1, t)…”

“…In this setting, Badr and Bernicot [5] defined a family of homogeneous atomic Hardy-Sobolev spacesḢS 1 t,ato and proved the following comparison between these spaces: Theorem 1.4. (See [5].) Let M be a complete Riemannian manifold satisfying a doubling condition and a Poincaré inequality (P q ) for some q > 1.…”

“…(See[5].) Let M be a complete Riemannian manifold satisfying (D) and a Poincaré inequality (P q ) for some q > 1.…”