Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0210137

A positive radial product formula for the Dunkl kernel

Margit Rösler
View original source ↗Read source document ↗Export metadata ↓

Abstract

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for non-negative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial hereby means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.

Explore related subjects

math.CAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Margit Rösler. 2002-10-09. A positive radial product formula for the Dunkl kernel. https://arxiv.org/abs/math/0210137

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Weighted inequalities in ergodic theory via transference

We first extend Calderón's transfer principle to weighted spaces in various different settings under suitable assumptions. Then we apply our results for some inequalities on the real line obtained by the author to prove corresponding inequalities in ergodic theory and ergodic $H^1$ spaces as well.

math.CA↗
arXiv2026

Wavelet resolution and Sobolev regularity of Calderón-Zygmund operators on domains

Given a uniform domain $Ω\subset {\mathbb R}^d$, we resolve each element of a suitably defined class of Calderòn-Zygmund (CZ) singular integrals on $Ω$ as the linear combination of Triebel wavelet operators and paraproduct terms. Our resolution formula entails a testing type characterization, loosely in the vein of the David-Journé theorem, of weighted Sobolev space bounds in terms of Triebel-Lizorkin and tree Carleson measure norms of the paraproduct symbols, which is new already in the case $Ω={\mathbb R}^d$ with Lebesgue measure. Our characterization covers the case of compressions to $Ω$ of global CZ operators, extending and sharpening past results of Prats and Tolsa for the convolution case. The weighted estimates we obtain, particularized to the Beurling operator on a Lipschitz domain with normal to the boundary in the corresponding sharp Besov class, may be used to deduce quantitative estimates for quasiregular mappings with dilatation in the Sobolev space $W^{1,p}(Ω)$, $p>2$.

math.CA↗