Sharp Hardy–Littlewood–Sobolev Inequality on the Upper Half Space

Abstract

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.

Subcritical approach to sharp Hardy–Littlewood–Sobolev type inequalities on the upper half space

“…HLS type inequalities essentially are the L p estimates for the convolution operators with this kernel, either in the whole space (the classical HLS inequality), or on the upper half space (Dou and Zhu's generalization of HLS inequality on the upper half space [5]). Note that this kernel, up to a constant, can be viewed as the fundamental solution of (−∆) α/2 operator.…”

“…Passing to the limit, we obtain the sharp inequality for p = 2(n−1) n+α−2 and t = 2n n+α . The extremal functions for the sharp inequality in critical case can be classified via the method of moving spheres (following from similar argument in Li [11] and Dou and Zhu [5]). In Section 3, we prove Theorem 1.2 by similar procedures as in Section 2.…”

“…Sharp inequality in subcritical case. For f (y) defined on ∂B n , we introduce the extension operator as in [5]:…”

“…As in the subcritical case (see Lemma 2.7), u(y), y ∈ ∂R n + is differentiable in y i , i = 1, 2, · · · , n − 1, and v(x), x ∈ R n + is differentiable in x i , i = 1, 2, · · · , n. Using the method of moving spheres, (see similar argument given in [4]- [5]), we know…”

“…Dou and Zhu [5] showed that: for 0 < α < n, 1 < p, t < ∞, n−1 n · 1 p + 1 t + n−α+1 n = 2, there is a constant C(p, n, α), such that, for any f ∈ L p (∂R…”