We give a new characterization of (homogeneous) Triebel-Lizorkin spaceṡ F s p,q (Z) in the smoothness range 0 < s < 1 for a fairly general class of metric measure spaces Z. The characterization uses Gromov hyperbolic fillings of Z. This gives a short proof of the quasisymmetric invariance of these spaces in case Z is Q-Ahlfors regular and sp = Q > 1. We also obtain first results on complex interpolation for these spaces in the framework of doubling metric measure spaces.

Abstract. The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well adapted to extending functions using the Whitney extension operator.

Abstract. For 0 < s < 1 < q < ∞, we characterize the homeomorphisms ϕ : R n → R n for which the composition operator f → f • ϕ is bounded on the homogeneous, scaling invariant Besov spaceḂ s n/s,q (R n ), where the emphasis is on the case q = n/s, left open in the previous literature. We also establish an analogous result for Besovtype function spaces on a wide class of metric measure spaces as well, and make some new remarks considering the scaling invariant Triebel-Lizorkin spacesḞ s n/s,q (R n ) with 0 < s < 1 and 0 < q ≤ ∞.

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.

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