Atomic decomposition on Hardy–Sobolev spaces

Abstract

Abstract. As a natural extension of L p Sobolev spaces, we consider Hardy-Sobolev spaces and establish an atomic decomposition theorem, analogous to the atomic decomposition characterization of Hardy spaces. As an application, we deduce several embedding results for Hardy-Sobolev spaces.

Abstract Hardy–Sobolev spaces and interpolation

“…For 1 < β ∞, we say that a function b is a homogeneous (1, β)-atom associated to a ball Q if (1) b is supported in the ball Q, (1, β)-atoms such that f = i λ i b i with i |λ i | < ∞. We equip this space with the semi-norm [33,16] in the Euclidean case). …”

“…We recall that R. Coifman proved an atomic decomposition for the classical Hardy space H 1 CW , which can be defined by maximal functions (see [23]). In the Euclidean case, the question of atomic decomposition for the homogeneous spaceḢS 1 was treated in [33] and [16]. However, in the non-Euclidean case this issue is still not clear.…”

“…We mention R. Strichartz [33]. Related works are [6,28,16,30]. They deal with "classical" Hardy-Sobolev spaces HS 1 on R n , which correspond to the Sobolev version of the Coifman-Weiss Hardy space H 1 CW (R n ): HS 1 is the set of functions f ∈ H 1 CW such that each partial derivative of f belongs to H 1 CW .…”

An atomic decomposition of the Hajłasz Sobolev spaceM11on manifolds

“…Further characterizations of HardySobolev spaces on R n by means of atoms are given in [8] and [25]. For related work see [20].…”

Hardy-Sobolev Spaces Associated with Twisted Convolution

“…In [12], the authors gave the atomic decomposition of the Hardy-Sobolev space and proved the endpoint case of the div-curl theorem of [13]. Also the papers of Cho and Kim [14], Janson [15], and Orobitg [16] are related to the theme of the present paper. Recently, functional spaces associated with operators are considered by more and more mathematicians.…”