Mathematics – Group Theory
Scientific paper
2008-10-22
Mathematics
Group Theory
PhD Thesis 2004
Scientific paper
The aim of my PhD work is to study the $L^p$-boundedness of operators on two classes of two-step nilpotent Lie groups, using Plancherel formulas and spherical functions as tools. The first class of groups consists of the groups of Heisenberg type, and the second, of the two-step free nilpotent Lie groups (denoted $N_{v,2}$ for $v$ generators). In the latter case, we develop a radial Fourier calculus. Our study has focused on the maximal functions associated with Kor\'anyi spheres, together with their square functions, and the convolution operator defined with the radial Fourier calculus on the two-step free nilpotent Lie group (radial Fourier multipliers problem). In fact, one chapter of this work is devoted to the proof of $L^p$-inequalities for the maximal spherical function on the two considered classes of groups. Our method is based on interpolation for the same operator family as in the euclidean case, on $L^p$-boundedness for the standard maximal function, and $L^2$-inequalities for square functions. These $L^2$-inequalities are based on Plancherel formula and on the properties of bounded spherical functions for the orthogonal group. On $N_{v,2}$, we construct the bounded spherical functions using representations of the semidirect product of $N_{v,2}$ with the orthogonal group. We also obtain some properties of the Kohn sublaplacian and the radial Plancherel measure. Then we present a first study of the radial Fourier multiplier problem, with the aim of giving our solutions for some technicals difficulties.
No associations
LandOfFree
Etude de deux classes de groupes nilpotents de pas deux does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Etude de deux classes de groupes nilpotents de pas deux, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Etude de deux classes de groupes nilpotents de pas deux will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-679958