The restrictions B s pq and F s pq of the Besov and Triebel--Lizorkin spaces of tempered distributions B s pq (R n ) and F s pq (R n ) to Lipschitz domains\Omega ae R n are studied. For general values of parameters (s 2 R, p ? 0, q ? 0) a "universal" linear bounded extension operator from B s pq and F s pq into the corresponding spaces on R n is constructed. The construction is based on a new variant of the Calder'on reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of B s pq and F s pq in terms of their values in\Omega are also obtained. Introduction The purpose of this paper is to construct a linear operator E which extends functions and distributions from a given Lipschitz domain\Omega ae R n to all of R n and possesses the following property: If a distribution f 2 D 0(\Omega\Gamma can be somehow extended to a tempered distribution g on R n which is "regular" in the sense that g 2 X for some function or...

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We deal with decay and boundedness properties of radial functions belonging to Besov and Lizorkin-Triebel spaces. In detail we investigate the surprising interplay of regularity and decay. Our tools are atomic decompositions in combination with trace theorems. 1

Atomic decompositions of function spaces