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3 Królikowski W.

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On Clifford-type structures

100%

Królikowski W.

Instytut Matematyczny Polskiej Akademii Nauk

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 $S^{r̅}_{p,q}A$) if, and only if, it can be decomposed as $f(x) = ∑_{ν}∑_{m} λ_{νm}a_{νm}(x)$, convergence in S', with coefficients $λ = {λ_{νm}}$ in a corresponding sequence space (say $s^{r̅}_{p,q}a$). These decomposition theorems establish a very useful connection between function and sequence spaces. We use them in the study of the decay of entropy numbers of compact embeddings between two function spaces of dominating mixed smoothness, reducing this problem to the same question on the sequence space level. The scales considered cover many important specific spaces (Sobolev, Zygmund, Besov) and we get generalisations of respective assertions of Belinsky, Dinh Dung and Temlyakov.

Fueter regular mappings and harmonicity

93%

Królikowski W.

nr 2

97-114

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.

Pairs of Clifford algebras of the Hurwitz type

93%

Królikowski W.

tom 37

nr 1

327-330

For a given Hurwitz pair $[S(Q_{S}),V(Q_{V}),o]$ the existence of a bilinear mapping $⭑: C(Q_{S}) × C(Q_{V}) → C(Q_{V})$ (where $C(Q_{S})$ 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.