Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows, $$left| {int_{mathbb{R}^n } {int_{mathbb{R}^n } {frac{{f(x)g(y)}} {{left| x right|^alpha x - yleft| {^lambda } right|left. y right|^beta }}dxdy} } } right| leqslant B(p,q,alpha ,lambda ,beta ,n)left| f right|_{L^p (mathbb{R}^n )} left| g right|_{L^q (mathbb{R}^n )} .$$ 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 $tfrac{1} {p} + tfrac{1} {q} > 1$ .