A sharp version of the Forelli-Rudin type estimates on the unit real ball (Q2790819)

The authors offer sharp Forelli-Rudin type estimates for the integrals NEWLINE\[NEWLINEI_s(x)= \int_{S^{n-1}} {d\sigma(\zeta)\over |x-\zeta|^{n-1+s}}\quad\text{and}\quad J_{s,t}(x)= \int_{B^n} {1-|y^2|^d)\,dV(y)\over [[x,y]]^{n+t+s}},NEWLINE\]NEWLINE where \(s\in\mathbb{R}\), \(t>-1\), \(x\in B_n\), which is the unit ball of \(\mathbb{R}^n\), and \(S^{n-1}\) its boundary. Here, \(dV\) is the Lebesgue measure on \(\mathbb{R}^n\), normalized so that \(V(B_n)= 1\), and \(d\sigma\) the surface measure on \(S^{-1}\) so that \(\sigma(S^{n-1})=1\). The Ahlfors bracket \([[x,y]]\) is defined by \([[x,y]]=)1-2x\cdot y+|x|^2\cdot|y|^2)^{1/2}\).NEWLINENEWLINE For example, if \(-1\leq s<0\) or \(s\leq 1-n\), then for all \(x\in B_n\) he has NEWLINE\[NEWLINE1\leq I_s(x)\leq (\Gamma(n/2)\cdot\Gamma(-s))/\Biggl[\Gamma\Biggl({1-s\over 2}\Biggr)\cdot\Gamma\Biggl({n-1-s\over 2}\Biggr)\Biggr].NEWLINE\]