Hardy—Littlewood—Sobolev Inequalities with the Fractional Poisson Kernel and Their Applications in PDEs

The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel on the upper half space ∫R+n∫∂R+nf(ξ)P(x,ξ,α)g(x)dξdx≤Cn,α,p,q′∥g∥Lq′(R+n)∥f∥Lq′(∂R+n), where f∈Lp(∂R+n),g∈Lq′(R+n)andp,q′∈(1+∞),2≤α<nsatisfyingn−1np+1q′+2−αn=1.Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C _n,α,p,q′ . Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore, in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler-Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ0∈∂R+n. Our results proved in this paper play a crucial role in establishing the Stein-Weiss inequalities with the Poisson kernel in our subsequent paper.