Topics on Harmonic analysis and Multilinear Algebra

Doctoral thesis, 2015

The present thesis consists of six different papers. Indeed, they treat three different research areas: function spaces, singular integrals and multilinear algebra. In paper I, a characterization of continuity of the $p$-$\Lambda$-variation function is given and Helly's selection principle for $\Lambda BV^{(p)}$ functions is established. A characterization of the inclusion of Waterman-Shiba classes into classes of functions with given integral modulus of continuity is given. A useful estimate on the modulus of variation of functions of class $\Lambda BV^{(p)}$ is found. In paper II, a characterization of the inclusion of Waterman-Shiba classes into $H_{\omega}^{q}$ is given. This corrects and extends an earlier result of a paper from 2005. In paper III, the characterization of the inclusion of Waterman-Shiba spaces $\:\Lambda BV^{(p)}\:$ into generalized Wiener classes of functions $BV(q;\,\delta)$ is given. It uses a new and shorter proof and extends an earlier result of U. Goginava. In paper IV, we discuss the existence of an orthogonal basis consisting of decomposable vectors for all symmetry classes of tensors associated with Semi-dihedral groups $SD_{8n}$. In paper V, we discuss o-bases of symmetry classes of tensors associated with the irreducible Brauer characters of the Dicyclic and Semi-dihedral groups. As in the case of Dihedral groups [46], it is possible that $V_\phi(G)$ has no o-basis when $\phi$ is a linear Brauer character. Let $\vec{P}=(p_1,\dotsc,p_m)$ with $1

Brauer symmetry classes of tensors

Lerner's formula

Helly's theorem

Modulus of variation

Generalized bounded variation

local mean oscillation

Orthogonal basis

weighted norm inequalities

weighted bounds

Multilinear singular integrals

Generalized Wiener classes

Symmetry classes of tensors