Abstract
On \(\mathbb {R}^N\) equipped with a normalized root system R and a multiplicity function \(k\ge 0\) let us consider a (not necessarily radial) kernel \(K({\mathbf {x}})\) satisfying \(|\partial ^\beta K({\mathbf {x}})|\lesssim \Vert {\mathbf {x}}\Vert ^{-{\mathbf {N}}-|\beta |}\) for \(|\beta |\le s\), where \({\mathbf {N}}\) is the homogeneous dimension of the system \(({\mathbb {R}}^N,R,k)\). We additionally assume that
where dw is the associated measure. We prove that if s large enough then a singular integral Dunkl convolution operator associated with the kernel \(K({\mathbf {x}})\) is bounded on \(L^p(dw)\) for \(1<p<\infty \) and of weak-type (1,1). Furthermore, we study a maximal function related to the Dunkl convolutions with truncation of K.
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Anker, J.-P., Dziubański, J., Hejna, A.: Harmonic functions, conjugate harmonic functions and the Hardy space \(H^1\) in the rational Dunkl setting. J. Fourier Anal. Appl. 25, 2356–2418 (2019)
Amri, B., Hammi, A.: Dunkl-Schrödinger operators, Complex Anal. Oper. Theory (2018)
Amri, B., Sifi, M.: Singular integral operators in the Dunkl setting. J. Lie Theory 22, 723–739 (2012)
de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113, 147–162 (1993)
de Jeu, M., Rösler, M.: Asymptotic analysis for the Dunkl kernel. J. Approx. Theory 119(1), 110–126 (2002)
Dunkl, C.F.: Reflection groups and orthogonal polynomials on the sphere. Math. Z. 197(1), 33–60 (1988)
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. 311(1), 167–183 (1989)
Dunkl, C.F.: Hankel transforms associated to finite reflection groups, in: Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications, Proceedings, Tampa 1991. Contemp. Math. 138, 123–138 (1989)
Dunkl, C.F.: Integral kernels with reflection group invariance. Canad. J. Math. 43(6), 1213–1227 (1991)
Duoandikoetxea, J.: Fourier Analysis, Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, (2001)
Dziubański, J., Hejna, A.: Remark on atomic decompositions for Hardy space \(H^1\) in the rational Dunkl setting. Studia Math. 251, 89–110 (2020)
Dziubański, J., Hejna, A.: Hörmander’s multiplier theorem for the Dunkl transform. J. Funct. Anal. 277, 2133–2159 (2019)
Gallardo, L., Rejeb, C.: A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications. Trans. Amer. Math. Soc. 368(5), 3727–3753 (2015)
Grafakos, L.: Modern Fourier Analysis, 3rd edition, Graduate Texts in Mathematics, 250. Springer, New York (2014)
Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192, 519–542 (1998)
Rösler, M.: Positivity of Dunkl’s intertwining operator. Duke Math. J. 98(3), 445–463 (1999)
Rösler, M.: A positive radial product formula for the Dunkl kernel. Trans. Amer. Math. Soc. 355(6), 2413–2438 (2003)
Rösler, M.: Dunkl operators (theory and applications). In: Koelink, E., Van Assche, W. (eds.) Orthogonal polynomials and special functions (Leuven, 2002), 93–135. Lect. Notes Math. 1817, Springer-Verlag (2003)
Rösler, M., Voit, M.: Dunkl theory, convolution algebras, and related Markov processes, in Harmonic and stochastic analysis of Dunkl processes, P. Graczyk, M. Rösler, M. Yor (eds.), 1–112, Travaux en cours 71, Hermann, Paris, (2008)
Stein, E.M.: Singular integral and differentiability properties of functions, Princeton Math. Series 30, Princeton University Press, New Jersy (1970)
Stein, E.M.: Harmonic analysis (real variable methods, orthogonality and oscillatory integrals), Princeton Math. Series 43, Princeton University Press, (1993)
Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005)
Thangavelu, S., Xu, Y.: Riesz transforms and Riesz potentials for the Dunkl transform. J. Comp. Appl. Math. 199, 181–195 (2007)
Triméche, K.: Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators. Integral Transforms Spec. Funct. 13(1), 17–38 (2002)
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Research supported by the National Science Centre, Poland (Narodowe Centrum Nauki), Grant 2017/25/B/ST1/00599.
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Dziubański, J., Hejna, A. Singular integrals in the rational Dunkl setting. Rev Mat Complut 35, 711–737 (2022). https://doi.org/10.1007/s13163-021-00402-1
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DOI: https://doi.org/10.1007/s13163-021-00402-1