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Sharp Constants in the HardyLittlewoodSobolev and Related Inequalities
Elliott H. Lieb
Annals of Mathematics
Second Series, Vol. 118, No. 2 (Sep., 1983), pp. 349374
Published by: Annals of Mathematics
DOI: 10.2307/2007032
Stable URL: http://www.jstor.org/stable/2007032
Page Count: 26
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Topics: Mathematical inequalities, Mathematical integrals, Mathematical constants, Mathematical functions, Fourier transformations, Mathematical theorems, Mathematical problems
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Abstract
A maximizing function, f, is shown to exist for the HLS inequality on Rn:  xλ * fq ≤ Np, λ, nfp, with N being the sharp constant and $1/p + \lambda/n = 1 + 1/q, 1 < p, q, n/ \lambda < \infty$. When p = q' or p = 2 or q = 2, f and N are explicitly evaluated. A maximizing f is also shown to exist for other inequalities: (i) The Okikiolu, Glaser, Martin, Grosse, Thirring inequality: $K_{n,p}\\nabla f\_2 \geq \ x^{b}f\_p, n \geq 3, 0 \leq b < 1, p = 2n/(2b + n  2)$. (This was known before, but the proof here has certain simplifications.) (ii) The doubly weighted HLS inequality of Stein and Weis: ∫ V (x, y)f(y) dyq ≤ Pα, β, p, λ, nfp with $V(x, y) = x^{\beta}x  y^{\lambda}y^{\alpha}, 0 \leq \alpha < n/p', 0 \leq \beta < n/q, 1/p + (\lambda + \alpha + \beta)/n = 1 + 1/q$. (iii) The weighted Young inequality:  xγ fm p ≥ Qp, m, n f(m)∞, where f(m)(x) is the mfold convolution of f with itself, $m \geq 3, m/( 1) \leq p < m, \gamma/n + 1/p = (m  1)/m$. When p = m/(m  1) or p = 2, f and Q are explicitly evaluated.
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Annals of Mathematics © 1983 Annals of Mathematics