On an integral inequality and application to Poisson’s equation

Let n ∈ N , m ± : = max / min { 1 , 8 n + 4 } and let I ( x ) = ⨍ B n ( 1 − | y | 2 ) ( | x − y | | y | | x − y ∗ | ) − n / 2 d y , where B n be the unit ball in R n . It is proved the double sharp inequality m − ⩽ I n ( x ) ⩽ m + . As an application, we obtain the following: if u is a solution to homogeneous Dirichlet’s problem of Poisson’s equation Δ u = g , g ∈ L ∞ , in the unit disk B 2 , then there holds the inequality | ∇ u | ⩽ 2 3 | g | ∞ .