We study several techniques which are well known in the case of Besov and Triebel-Lizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal with so-called atomic, subatomic and wavelet decompositions. All these theorems have much in common. Roughly speaking, they say that a function f belongs to some function space (say ) if, and only if, it can be decomposed as
, convergence in S’,
with coefficients...
We study Sobolev-type embeddings involving rearrangement-invariant norms. In particular, we focus on the question when such embeddings are optimal. We concentrate on the case when the functions involved are defined on R. This subject has been studied before, but only on bounded domains. We first establish the equivalence of the Sobolev embedding to a new type of inequality involving two integral operators. Next, we show this inequality to be equivalent to the boundedness of a certain Hardy operator...
We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in , with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to . The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.
We continue our earlier investigations of radial subspaces of Besov and Lizorkin-Triebel spaces on . This time we study characterizations of these subspaces by differences.
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