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This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.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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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