On an integral inequality and application to Poisson’s equation

Let n∈Nn∈N, m±:=max/min{1,8n+4} and let I(x)=⨍Bn(1−|y|2)(|x−y||y||x−y∗|)−n/2dy, where BnBn be the unit ball in RnRn. It is proved the double sharp inequality m−⩽In(x)⩽m+m−⩽In(x)⩽m+. As an application, we obtain the following: if uu is a solution to homogeneous Dirichlet’s problem of Poisson’s equation Δu=g, g∈L∞g∈L∞, in the unit disk B2B2, then there holds the inequality |∇u|⩽23|g|∞.