Abstract
In this paper we investigate Riesz transforms R μ (k) of order k≥1 related to the Bessel operator Δμ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R μ (k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x 2μ+1 dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R μ (k) maps L p(ω) into itself and L 1(ω) into L 1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class \(\mathcal{A}_{p}^\mu\) of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.
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Altenburg, G., Bessel-Transformationen in Räumen von Grundfunktionen über dem Intervall Ω=(0,∞) und deren Dualräumen, Math. Nachr. 108 (1982), 197–218.
Andersen, K. F. and Kerman, R. A., Weighted norm inequalities for generalized Hankel conjugate transformations, Studia Math. 71 (1981/82), 15–26.
Andersen, K. F. and Muckenhoupt, B., Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9–26.
Betancor, J. J., Fariña, J. C., Buraczewski, D., Martínez, T. and Torrea, J. L., Riesz transforms related to Bessel operators, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 701–725.
Graczyk, P., Loeb, J.-J., López, P., Ins, A., Nowak, A., Urbina R. and Wilfredo, O., Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, J. Math. Pures Appl. 84 (2005), 375–405.
Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, Academic Press, New York, 1965.
Gutiérrez, C. E., Segovia, C. and Torrea, J. L., On higher Riesz transforms for Gaussian measures, J. Fourier Anal. Appl. 2 (1996), 583–596.
Haimo, D. T., Integral equations associated with Hankel convolutions, Trans. Amer. Math. Soc. 116 (1965), 330–375.
Hirschman, I. I. Jr., Variation diminishing Hankel transforms, J. Anal. Math. 8 (1960/1961), 307–336.
Kerman, R. A., Generalized Hankel conjugate transformations on rearrangement invariant spaces, Trans. Amer. Math. Soc. 232 (1977), 111–130.
Kerman, R. A., Boundedness criteria for generalized Hankel conjugate transformations, Canad. J. Math. 30 (1978), 147–153.
Muckenhoupt, B., Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38.
Muckenhoupt, B. and Stein, E. M., Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92.
Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30, Princeton University Press, Princeton, NJ, 1970.
Stein, E. M., Topics in Harmonic Analysis Related to the Littlewood–Paley Theory, Annals of Mathematics Studies 63, Princeton University Press, Princeton, NJ, 1970.
Stempak, K., The Littlewood–Paley theory for the Fourier–Bessel transform, Preprint 45, Math. Inst. Univ. Wrocław, 1985.
Weinstein, A., Discontinuous integrals and generalized potential theory, Trans. Amer. Math. Soc. 63 (1948), 342–354.
Zemanian, A. H., Generalized Integral Transformations, Dover, New York, 1987.
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Betancor, J., Fariña, J., Martinez, T. et al. Higher order Riesz transforms associated with Bessel operators. Ark Mat 46, 219–250 (2008). https://doi.org/10.1007/s11512-008-0078-9
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DOI: https://doi.org/10.1007/s11512-008-0078-9