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We discuss in which cases the Dunkl convolution u∗kv$u*_kv$ of distributions u,v$u,v$, possibly both with non‐compact support, can be defined and study its analytic properties. We prove results on the (singular‐)support of Dunkl convolutions. Based on this, we are able to prove a theorem on elliptic regularity for a certain class of Dunkl operators, called elliptic Dunkl operators. Finally, for the root system An−1$A_{n-1}$ we consider the Riesz distributions (Rα)α∈C$(R_\alpha)_{\alpha \in \mathbb {C}}$ and prove that their Dunkl convolution exists and that Rα∗kRβ=Rα+β$R_\alpha *_kR_\beta = R_{\alpha +\beta }$ holds. [ABSTRACT FROM AUTHOR]

The Dunkl operator as an object of mathematical physics is considered, we study the kernel and the surjectivity of Dunkl convolution operators in the space of entire functions and the space of entire functions of exponential type. The main result is the solution of the multipoint de la Vallee–Poussin problem for Dunkl convolution operators in the space of entire functions.

We introduce first weighted function spaces on $${\mathbb{R}^d}$$ using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on $${\mathbb{R}^d}$$ weighted L-estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L when this function belongs to a suitable Besov-Dunkl space. [ABSTRACT FROM AUTHOR]

In this paper, we establish on ℝd that the Dunkl transform maps the Besov spaces into a class of Herz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we show the inclusion and the density of the Schwartz space in Besov-Dunkl spaces and we prove an interpolation formula for these spaces by the real method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable Besov-Dunkl space.