Plewa, Paweł Sharp Hardy’s type inequality for Laguerre expansions. (English) Zbl 1509.42037 J. Math. Soc. Japan 74, No. 2, 333-352 (2022). Summary: A method of proving Hardy’s type inequality for orthogonal expansions is presented in a rather general setting. Then, sharp multi-dimensional Hardy’s inequality associated with the Laguerre functions of convolution type is proved for the type index \(\alpha \in [-1/2, \infty)^d\). The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy’s type inequality involving \(L^1\) norms in place of \(H^1\) norms are obtained in both settings. Cited in 1 Document MSC: 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42B30 \(H^p\)-spaces Keywords:Hardy’s inequality; Hardy’s space; Laguerre expansions of convolution type; standard Laguerre expansions PDFBibTeX XMLCite \textit{P. Plewa}, J. Math. Soc. Japan 74, No. 2, 333--352 (2022; Zbl 1509.42037) Full Text: DOI arXiv References: [1] J.-P. Anker, N. Ben Salem, J. Dziubański and N. Hamda, The Hardy space \(H^1\) in the rational Dunkl setting, Constr. Approx., 42 (2015), 93-128. · Zbl 1320.42013 [2] R. Askey, A transplantation theorem for Jacobi coefficients, Pacific J. Math., 21 (1967), 393-404. · Zbl 0172.08601 [3] R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math., 87 (1965), 695-708. · Zbl 0125.31301 [4] R. Balasubramanian and R. Radha, Hardy-type inequalities for Hermite expansions, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), no. 1, article 12, 4pp. · Zbl 1087.42021 [5] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. · Zbl 0358.30023 [6] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, Math. Annal., 97 (1927), 159-209. · JFM 52.0267.01 [7] Y. Kanjin, Hardy’s inequalities for Hermite and Laguerre expansions, Bull. London Math. Soc., 29 (1997), 331-337. · Zbl 0879.42019 [8] Y. Kanjin, Hardy’s inequalities for Hermite and Laguerre expansions revisited, J. Math. Soc. Japan, 63 (2011), 753-767. · Zbl 1225.42020 [9] Y. Kanjin and K. Sato, Hardy’s inequality for Jacobi expansions, Math. Inequal. Appl., 7 (2004), 551-555. · Zbl 1070.42020 [10] N. N. Lebedev, Special Functions and Their Applications, Dover Publ., Inc., New York, 1972. · Zbl 0271.33001 [11] Z. Li, Y. Yu and Y. Shi, The Hardy inequality for Hermite expansions, J. Fourier Anal. Appl., 21 (2015), 267-280. · Zbl 1325.42035 [12] B. Muckenhoupt, Mean convergence of Hermite and Laguerre series. II, Trans. Amer. Math. Soc., 147 (1970), 433-460. · Zbl 0191.07602 [13] I. Nåsell, Rational bounds for ratios of modified Bessel functions, SIAM J. Math. Anal., 9 (1978), 1-11. · Zbl 0377.33010 [14] A. Nowak and K. Stempak, On \(L^p\)-contractivity of Laguerre semigroups, Illinois J. Math., 56 (2012), 433-452. · Zbl 1279.47062 [15] P. Plewa, Hardy’s inequality for Laguerre expansions of Hermite type, J. Fourier Anal. Appl., 25 (2019), 1855-1873. · Zbl 1417.42030 [16] P. Plewa, On Hardy’s inequality for Hermite expansions, Taiwanese J. Math., 24 (2020), 301-315. · Zbl 1437.42043 [17] P. Plewa, Sharp Hardy’s inequality for Jacobi and symmetrized Jacobi trigonometric expansions, J. Approx. Theory., 256 (2020), 105422. · Zbl 1440.42135 [18] R. Radha, Hardy-type inequalities, Taiwanese J. Math., 4 (2000), 447-456. · Zbl 0998.42016 [19] R. Radha and S. Thangavelu, Hardy’s inequalities for Hermite and Laguerre expansions, Proc. Amer. Math. Soc., 132 (2004), 3525-3536. · Zbl 1062.42021 [20] M. Satake, Hardy’s inequalities for Laguerre expansions, J. Math. Soc. Japan, 52 (2000), 17-24. · Zbl 0946.42020 [21] K. Stempak, Heat-diffusion and Poisson integrals for Laguerre expansions, Tôhoku Math. J. (2), 46 (1994), 83-104. · Zbl 0793.42019 [22] G. Szegő, Orthogonal Polynomials, fourth edition, Amer. Math. Soc. Colloq. Publ., 23, Providence, 1975. · Zbl 0305.42011 [23] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, 42, Princeton Univ. Press, Princeton, 1993 · Zbl 0791.41030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.