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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 $S_{p,q}^{r̅}A$) if, and only if, it can be decomposed as $f\left(x\right)={\sum }_{\nu }{\sum }_m{\lambda }_{\nu m}{a}_{\nu m}\left(x\right)$, convergence in S’, with coefficients...

It is shown that Fueter regular functions appear in connection with the Eells condition for harmonicity. New conditions for mappings from 4-dimensional conformally flat manifolds to be harmonic are obtained.

For a given Hurwitz pair $[S\left(Q_S\right),V\left(Q_V\right),o]$ the existence of a bilinear mapping $⭑:C\left(Q_S\right)\times C\left(Q_V\right)\to C\left(Q_V\right)$ (where $C\left(Q_S\right)$ and $C(Q_V$) denote the Clifford algebras of the quadratic forms $Q_S$ and $Q_V$, respectively) generated by the Hurwitz multiplication “o” is proved and the counterpart of the Hurwitz condition on the Clifford algebra level is found. Moreover, a necessary and sufficient condition for "⭑" to be generated by the Hurwitz multiplication is shown.

Using the fundamental notions of the quaternionic analysis we show that there are no 4-dimensional almost Kähler manifolds which are locally conformally flat with a metric of a special form.

CONTENTSIntroduction........................................................................................................6I. Quaternionic regular and biregular functions in the sense of Fueter..............9   1. Introduction................................................................................................9   2. Fueter derivative and regular functions.....................................................10   3. Quaternionic partial derivatives.................................................................12   4....