Abstract
This paper presents a systematic study for harmonic analysis associated with the one-dimensional Dunkl transform, which is based upon the generalized Cauchy–Riemann equations D x u−∂ y v=0,∂ y u+D x v=0, where D x is the Dunkl operator (D x f)(x)=f′(x)+(λ/x)(f(x)−f(−x)). Various properties about the λ-subharmonic function, the λ-Poisson integral, the conjugate λ-Poisson integral, and the associated maximal functions are obtained, and the λ-Hilbert transform , a crucial analog to the classical one, is introduced and studied by a stringent method. The theory of the associated Hardy spaces \(H_{\lambda}^{p}({\mathbb{R}}^{2}_{+})\) on the half-plane \({\mathbb{R}}^{2}_{+}\) for p≥p 0=2λ/(2λ+1) with λ>0 extends the results of Muckenhoupt and Stein about the Hankel transform to a general case and contains a number of further results. In particular, the λ-Hilbert transform is shown to be a bounded mapping from \(H_{\lambda}^{1}({\mathbb{R}})\) to \(L^{1}_{\lambda}({\mathbb{R}})\); and associated to the Dunkl transform, an analog of the well-known Hardy inequality is proved for \(f\in H^{1}_{\lambda}({\mathbb{R}})\).
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References
Bers, L.: Function-theoretical properties of solutions of partial differential equations of elliptic type. In: Contributions to the Theory of Partial Differential Equations, Princeton, pp. 69–94 (1954)
Betancor, J.J., Ciaurri, Ó., Varona, J.L.: The multiplier of the interval [−1,1] for the Dunkl transform on the real line. J. Funct. Anal. 242, 327–336 (2007)
de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113, 147–162 (1993)
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)
Dunkl, C.F.: Integral kernels with reflection group invariance. Can. J. Math. 43, 1213–1227 (1991)
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, Tampa, 1991. Contemp. Math., vol. 138, pp. 123–138 (1992)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vols. I and II. McGraw-Hill, New York (1953)
Flett, T.M.: On the rate of growth of mean values of holomorphic and harmonic functions. Proc. Lond. Math. Soc. 20, 749–768 (1970)
Li, Zh.-K.: Hardy spaces for Jacobi expansions, I. The basic theory. Analysis 16, 27–49 (1996)
Li, Zh.-K., Liao, J.-Q.: Hardy spaces for Dunkl–Gegenbauer expansions. Preprint
Li, Zh.-K., Liu, L.-M., Song, F.-T.: Convolution operators for the Dunkl transform. Preprint
Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)
Rösler, M.: Bessel-type signed hypergroups on ℝ. In: Heyer, H., Mukherjea, A. (eds.) Probability Measures on Groups and Related Structures XI, pp. 292–304. World Scientific, Singapore (1995)
Soltani, F.: Littlewood–Paley operators associated with the Dunkl operator on ℝ. J. Funct. Anal. 221, 205–225 (2005)
Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables, I. The theory of H p-spaces. Acta Math. 103, 25–62 (1960)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton (1971)
Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005)
Zygmund, A.: Trigonometric Series, vols. I and II, 2nd edn. Cambridge Univ. Press, Cambridge (1959)
Acknowledgements
This research was supported by the National Natural Science Foundation of China (No. 10971141) and the Beijing Natural Science Foundation (No. 1122011).
The authors would like to thank the anonymous referees for their helpful comments and suggestions which have improved the original manuscript.
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Communicated by Doron S. Lubinsky.
Dedicated to Professor Lizhong Peng on the occasion of his 70th birthday.
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Li, Z., Liao, J. Harmonic Analysis Associated with the One-Dimensional Dunkl Transform. Constr Approx 37, 233–281 (2013). https://doi.org/10.1007/s00365-013-9179-1
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DOI: https://doi.org/10.1007/s00365-013-9179-1