## Taiwanese Journal of Mathematics

### On Hardy's Inequality for Hermite Expansions

Paweł Plewa

#### Abstract

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

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

Dates
Accepted: 2 June 2019
First available in Project Euclid: 5 June 2019

https://projecteuclid.org/euclid.twjm/1559700015

Digital Object Identifier
doi:10.11650/tjm/190601

Mathematical Reviews number (MathSciNet)
MR4078199

Zentralblatt MATH identifier
07192936

#### Citation

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

#### References

• 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.