Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

Abstract

Abstract:We establish trace theorems for function spaces de ned on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < , as well as the rst order Hajłasz-Sobolev space M ,p (Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces de ned intrinsically on F. Our method employs the de nitions of the function spaces via hyperbolic llings of the underlying metric space.

Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings

“…(ii) For some applications (see [23]), it is also useful to note that our methods allow some flexibility in the choice of the balls associated with the hyperbolic filling. More precisely, the parameters could be chosen so that (ξ x ) x∈Xn is for all n a set of points in Z with pairwise distances ≥ c 1 2 −n for some fixed constant c 1 (independent of n), that the radii r x corresponding to the balls B(x) := B(ξ x , r x ) (x ∈ X n ) are comparable to 2 −n uniformly in x and n, and that the balls c 2 B(x) x∈Xn cover Z for all n where c 2 ∈ (0, 1) is a fixed constant.…”

“…The new definition is also amenable to complex interpolation results, as will be discussed below. Moreover, this approach in conjunction with a suitable retraction result (see Proposition 6.3 below) leads to new and general trace theorems for Sobolev, Besov and Triebel-Lizorkin spaces in the setting of Ahlfors regular metric spaces; this question will be studied in a separate paper [23].…”

Traces of weighted function spaces: Dyadic norms and Whitney extensions

“…which is of the same form as the quantity (1) in the introduction. (8) and (9) in the setting of metric measure spaces were also considered in [43,Definition 5.1] in terms of a hyperbolic filling of R d . Another similar variant in the weighted Euclidean setting has been considered in [54].…”

“…Works focusing on trace theorems for fractals and related subsets of a Euclidean space include [25,41,42,50,44,23,24,5,17] (we also refer to [55] for a recent result concerning traces on non-regular subsets of R d ), while trace theorems in more general metric settings have been considered e.g. in [14,43,28,31,30]. In fact, the characterizations of fractional smoothness spaces as retracts of certain sequence spaces in [12], [18,Section 7] and [4,Proposition 6.3] can also be seen as abstract trace theorems.…”

“…Finally, the terms O 3 Q are essentially symmetric to with the terms O 1 Q , so H 3 can be estimated using the same argument as H 1 . Combining these estimates with (44) and applying them to (43), we arrive at…”