Convolution operator and maximal function for the Dunkl transform

By S. Thangavelu, Yuan Xu. 2004

304 citations recorded by the source corpus.

For a family of weight functionshK invariant under a finite reflection group onRd, analysis related to the Dunkl transform is carried out for the weightedLp spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e0a4e8e12ea16054501b16bb281245dc1420dcdc

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In Applied Mathematics

1. Hatem Mejjaoli
2. S. Tikhonov
3. Jean-Philippe Anker
4. Marcel de Jeu
5. Dmitry V. Gorbachev
6. V. Ivanov
7. N. B. Andersen
8. Feng Dai

Citations in the corpus

• A transference theorem for the Dunkl transform and its applications (2010, methodology): “In fact, even the L boundedness of τy is not established in general (see [20, 19]).”
• Positive $$L^p$$Lp-Bounded Dunkl-Type Generalized Translation Operator and Its Applications (2017, methodology): “x) = Stf(x) = Z Sd−1 f(x+ty′)dσ 0(y ′). Theorem 3.1. If 1 ≤ p≤ ∞, then, for any t∈ Rand f∈ S(Rd), (3.6) kTtfk p,dµk ≤ kfkp,dµ k . Remark 3.1. (i) The inequality kTtfk p,dµk ≤ ckfkp,dµ k was proved in [49] for G= Zd 2. (ii) Since S(Rd) is dense in Lp(Rd,dµ k), 1 ≤ p<∞, then for any t∈ R+ the opera”
• Spectrum of Functions for the Dunkl Transform on R^d (2007, background): “So g ∗D ψ = 0 for all ψ in D(IRd), which implies that g = 0 (see [12],[16],[17] for the properties of the Dunkl convolution).”
• An Introduction to Dunkl Theory and Its Analytic Aspects (2016, background): “[41] (see also [66] and the references therein). ‚ Many current works deal with generalizations of results in Euclidean harmonic analysis to the rational Dunkl setting. Among others, let us mention ˝ [84] about the Hardy–Littlewood and the Poisson maximal functions, ˝ [6] and [7] about singular inte”
• Hörmander's multiplier theorem for the Dunkl transform (2018, unknown): “,[9], [10], [12], [13], [16], [18], [20], [19], [24], [23]).”
• Paley-Wiener theorems for the Dunkl transform (2004, background): “Recently, an alternative reduction to the one-dimensional case has been introduced which circumvents the inference of the general case from the radial one [35].”
• REAL PALEY―WIENER THEOREMS AND LOCAL SPECTRAL RADIUS FORMULAS (2008, background): “A context for a possible application is the following: In higher dimension, the results for complex Paley–Wiener theorems for the Dunkl transform in [43] and [60] seem to be not yet optimal, as the version for supports in compact convex invariant sets is thought to be true in general, but has only b”