Taiwanese Journal of Mathematics

On Hardy's Inequality for Hermite Expansions

Paweł Plewa

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Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\alpha \in [-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validates that the known version of Hardy's inequality for the Hermite functions is sharp.

Article information

Taiwanese J. Math., Volume 24, Number 2 (2020), 301-315.

Received: 17 January 2019
Accepted: 2 June 2019
First available in Project Euclid: 5 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Secondary: 42B30: $H^p$-spaces 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Hardy's inequality Hardy's space Hermite expansions Laguerre expansions of Hermite type


Plewa, Paweł. On Hardy's Inequality for Hermite Expansions. Taiwanese J. Math. 24 (2020), no. 2, 301--315. doi:10.11650/tjm/190601. https://projecteuclid.org/euclid.twjm/1559700015

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  • R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695–708.
  • R. Balasubramanian and R. Radha, Hardy-type inequalities for Hermite expansions, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 1, Article 12, 4 pp.
  • R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645.
  • G. H. Hardy and J. E. Littlewood, Some new properties of fourier constants, Math. Ann. 97 (1927), no. 1, 159–209.
  • Y. Kanjin, Hardy's inequalities for Hermite and Laguerre expansions, Bull. London Math. Soc. 29 (1997), no. 3, 331–337.
  • ––––, Hardy's inequalities for Hermite and Laguerre expansions revisited, J. Math. Soc. Japan 63 (2011), no. 3, 753–767.
  • Y. Kanjin and K. Sato, Hardy's inequality for Jacobi expansions, Math. Inequal. Appl. 7 (2004), no. 4, 551–555.
  • N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972.
  • Z. Li, Y. Yu and Y. Shi, The Hardy inequality for Hermite expansions, J. Fourier Anal. Appl. 21 (2015), no. 2, 267–280.
  • B. Muckenhoupt, Mean convergence of Hermite and Laguerre series II, Trans. Amer. Math. Soc. 147 (1970), 433–460.
  • I. Nåsell, Rational bounds for ratios of modified Bessel functions, SIAM J. Math. Anal. 9 (1978), no. 1, 1–11.
  • P. Plewa, Hardy's inequality for Laguerre expansions of Hermite type, Accepted in J. Fourier Anal. Appl. (2018), 19 pp.
  • ––––, Sharp Hardy's type inequality for Laguerre expansions, preprint (2018), arXiv:1810.08138.
  • R. Radha, Hardy-type inequalities, Taiwanese J. Math. 4 (2000), no. 3, 447–456.
  • R. Radha and S. Thangavelu, Hardy's inequalities for Hermite and Laguerre expansions, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3525–3536.
  • M. Satake, Hardy's inequalities for Laguerre expansions, J. Math. Soc. Japan 52 (2000), no. 1, 17–24.
  • E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993.
  • K. Stempak, Heat-diffusion and Poisson integrals for Laguerre expansions, Tohoku Math. J. (2) 46 (1994), no. 1, 83–104.
  • G. Szegö, Orthogonal polynomials, Fourth edition, American Mathematical Society, Colloquium Publication XXIII, American Mathematical Society, Providence, R.I., 1975.