Journal of the Mathematical Society of Japan

Hardy's inequalities for Hermite and Laguerre expansions revisited


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We show that Hardy's inequalities for Laguerre expansions hold on the space $L^1(0,\infty)$ when the Laguerre parameters $\alpha$ are positive, and we prove that although the inequality holds on the real Hardy space $H^1(0,\infty)$ if $\alpha= 0$, it does not hold on $L^1(0,\infty)$. Further, Hardy's inequality for Hermite expansion is established on $L^1(0,\infty)$.

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J. Math. Soc. Japan Volume 63, Number 3 (2011), 753-767.

First available in Project Euclid: 1 August 2011

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Mathematical Reviews number (MathSciNet)

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 Hermite expansion Laguerre expansion


KANJIN, Yuichi. Hardy's inequalities for Hermite and Laguerre expansions revisited. Journal of the Mathematical Society of Japan 63 (2011), no. 3, 753--767. doi:10.2969/jmsj/06330753.

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