Dunkl--BMO functions representable in terms of generalized balayage of Carleson measures and an application

Qingdong Guo

Home/Archive/arXiv

arXiv · 2610.05092

Dunkl--BMO functions representable in terms of generalized balayage of Carleson measures and an application

Qingdong Guo

View original source ↗Read source document ↗Export metadata ↓

Abstract

We characterize compactly supported functions $f$ in Dunkl--BMO spaces as those that can be written as \begin{equation*} f=g+\widetilde{S}_{μ,\mathcal{P}}, \end{equation*} the sum of some bounded function $g$ and the generalized balayage $\widetilde{S}_{μ,\mathcal{P}}$ of a Carleson measure on $\mathbb{R}^{N}\times(0,\infty)$ associated with the Dunkl--Poisson semigroup. As a direct application of this representation of Dunkl--BMO functions, we establish a new characterization of the Hardy space in the Dunkl setting via a principal value integral related to $\widetilde{S}_{μ,\mathcal{P}}$. Moreover, analogous conclusions hold for the Dunkl--heat semigroup as well.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Qingdong Guo. 2026-10-04. Dunkl--BMO functions representable in terms of generalized balayage of Carleson measures and an application. https://arxiv.org/abs/2610.05092

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

KEEP EXPLORING

Related papers

arXiv2026

A variational principle for Gaussian lattice sums

We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices $Λ\subset \mathbb{R}^2$ with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be seen as the inhomogeneous counterpart to Montgomery's 1988 result, which proved that the hexagonal lattice minimizes the maximum.

math.CA↗

arXiv2026

Dimension-free estimates for discrete maximal functions related to normalized sampled gaussians

In this paper, we investigate dimension-free estimates for convolution operators with discrete normalized sampled Gaussians (related to the Theta function) in the context of maximal, jump, $r$-variational and oscillation inequalities on $\ell^p(\mathbb{Z}^d)$ spaces. This is the first instance of a discrete maximal function in the literature - a non-semigroup example - where dimension-free $\ell^p(\mathbb{Z}^d)$ bounds are provided for the entire range of $1 < p < \infty$. The methods of proof rely on developing robust Fourier techniques, which are combined with the fractional derivative, a tool that has not been previously applied to studying similar questions in the discrete setting.

math.CA↗

arXiv2026

Characterizations of fractional operators via integral transforms

In 1974, J. S. Lew established a reasonable conjecture regarding an axiomatic characterization for the one-dimensional Riemann--Liouville integral. This conjecture was proved by Cartwright and McMullen in 1978. After that, little further work has been done on this topic, except some extensions for the Stieltjes case in one and several variables. In this paper, we prove the necessity of the axioms established in the conjecture of J. S. Lew using the Cauchy functional equation and Hamel bases. In addition, we give a proof for the characterization in several variables by employing the Titchmarsh theorem, as a natural extension of the approach of Cartwright and McMullen. We also provide an alternative version and proof in one and several variables with Laplace transforms and the Cauchy functional equation, weakening parts of the continuity assumption. We show a similar result for the Riesz potential in terms of the Fourier transform. Finally, we illustrate how the theory can be used for characterization in the context of fractional calculus with respect to a non-smooth integrator, based on transmutation and measures.

math.CA↗