Abstract
It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,
The main purpose of this paper is to give the sharp constants B(p, q, α, λ, β, n) for the above inequality for three cases: (i) p = 1 and q = 1; (ii) p = 1 and 1 < q ⩽ ∞, or 1 < p ⩽ ∞ and q = 1; (iii) 1 < p, q < ∞ and \(\tfrac{1} {p} + \tfrac{1} {q} = 1\). In addition, the explicit bounds can be obtained for the case 1 < p, q < ∞ and \(\tfrac{1} {p} + \tfrac{1} {q} > 1\).
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Wu, D., Shi, Z. & Yan, D. Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality. Sci. China Math. 57, 963–970 (2014). https://doi.org/10.1007/s11425-013-4681-2
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DOI: https://doi.org/10.1007/s11425-013-4681-2