Real interpolations for Besov and Triebel‐Lizorkin spaces on spaces of homogeneous type

Abstract

The author establishes a full real interpolation theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. The corresponding theorem for homogeneous Besov and Triebel-Lizorkin spaces is also presented. Moreover, as an application, the author gives the reduced T 1 theorem for homogeneous Besov spaces on spaces of homogeneous type.

Interpolation properties of Besov spaces defined on metric spaces

“…Our definition is rather concrete and gives the usual Besov space in the Euclidean setting. Moreover, it has very recently been shown by Müller and Yang [18] that our definition here coincides with the definition based on test functions and used earlier by Han [10], Han and Yang [11] and Yang [22], provided that X, besides being doubling, also satisfies a reverse doubling condition. We show in Section 5 that, under a p-Poincaré inequality assumption, the Besov space B α p,q (X), 0 < α < 1, is realized as the real interpolation space (L p (X), KS 1,p (X)) α,q between the corresponding L p (X) and Sobolev spaces.…”

“…Under the assumption that X be Ahlfors regular, the interpolation result Theorem 4.4 has been established by Yang [22]. An analogous version of the above interpolation theorem has been established by Han, Müller and Yang [8,Theorem 8.1 and Theorem 8.3] for Besov type spaces constructed using frames and approximations of the identity in general metric measure spaces equipped with a doubling measure that in addition has a reverse doubling property.…”

Measure Density and Extension of Besov and Triebel–Lizorkin Functions

“…[6] using different methods. For related interpolation results in the metric setting, see [46], [16] and [9].…”

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

“…This generalizes [18,Theorem 1.4] and [19,Theorem 4.1] by taking s ∈ (0, 1), p ∈ (max{n/(n + ), n/(n + + s)}, ∞] and q ∈ (0, ∞]. As an application of these coincidences and via the Calderón reproducing formulae in [15], we establish some real interpolation conclusions of the spaces AḂ s p,q (X ) and AḞ s p,q (X ), which generalize the real interpolation theorems of Besov and Triebel-Lizorkin spaces on Ahlfors n-regular metric spaces in [23] and RD-spaces in [15]; see Theorem 1.2 below. The corresponding results on inhomogeneous grand Besov spaces A B We begin with the notion of RD-spaces in [15] (see also [24]).…”