In this paper, we obtain a reversed Hardy-Littlewood-Sobolev inequality: for 0 < p, t < 1 and λ = n − α < 0 with 1/p + 1/t + λ/n = 2, there is a best constant N (n, λ, p) > 0, such thatholds for all nonnegative functions f ∈ L p (R n ), g ∈ L t (R n ). For p = t, we prove the existence of extremal functions, classify all extremal functions via the method of moving sphere, and compute the best constant.Mathematics Subject Classification(2010). 35A23, 42B37

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent λ = n − α (that is for the case of α > n). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.

Abstract. In this paper we establish the reversed sharp Hardy-LittlewoodSobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in [8]) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.

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