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Journal Article

# Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities

Elliott H. Lieb
Annals of Mathematics
Second Series, Vol. 118, No. 2 (Sep., 1983), pp. 349-374
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: | |x|-λ * f|q ≤ Np, λ, n|f|p, 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) dy|q ≤ Pα, β, p, λ, n|f|p 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: | |x|γ f|m p ≥ Qp, m, n| f(m)|∞, where f(m)(x) is the m-fold 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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