231,110 research outputs found
On fundamental harmonic analysis operators in certain Dunkl and Bessel settings
We consider several harmonic analysis operators in the multi-dimensional
context of the Dunkl Laplacian with the underlying group of reflections
isomorphic to (also negative values of the multiplicity
function are admitted). Our investigations include maximal operators,
-functions, Lusin area integrals, Riesz transforms and multipliers of
Laplace and Laplace-Stieltjes transform type. Using the general
Calder\'on-Zygmund theory we prove that these objects are bounded in weighted
spaces, , and from into weak .Comment: 26 pages. arXiv admin note: text overlap with arXiv:1011.3615 by
other author
On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings
We study several fundamental harmonic analysis operators in the
multi-dimensional context of the Dunkl harmonic oscillator and the underlying
group of reflections isomorphic to . Noteworthy, we admit
negative values of the multiplicity functions. Our investigations include
maximal operators, -functions, Lusin area integrals, Riesz transforms and
multipliers of Laplace and Laplace-Stieltjes type. By means of the general
Calder\'on-Zygmund theory we prove that these operators are bounded on weighted
spaces, , and from weighted to weighted weak .
We also obtain similar results for analogous set of operators in the closely
related multi-dimensional Laguerre-symmetrized framework. The latter emerges
from a symmetrization procedure proposed recently by the first two authors. As
a by-product of the main developments we get some new results in the
multi-dimensional Laguerre function setting of convolution type
An algorithm for the Cartan-Dieudonn\'e theorem on generalized scalar product spaces
We present an algorithmic proof of the Cartan-Dieudonn\'e theorem on
generalized real scalar product spaces with arbitrary signature. We use
Clifford algebras to compute the factorization of a given orthogonal
transformation as a product of reflections with respect to hyperplanes. The
relationship with the Cartan-Dieudonn\'e-Scherk theorem is also discussed in
relation to the minimum number of reflections required to decompose a given
orthogonal transformation.Comment: 25 page
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