Generalizations of the Forelli-Rudin estimates (Q1840551)

The author defines a Rudin-Forelli weight function to be an increasing differentiable function \(\eta :[0,1)\rightarrow [0,\infty)\) such that \((1-r)\eta '(r)=o(\eta (r))\) as \(r\rightarrow 1\). For such a function the following estimates are proved for \(B_{n}\) the unit ball in \(\mathbb{C}^{n}\) with volume element \(dV\): \[ \int_{B_{n}}\frac{(1-|z|^{2})^{t}}{|1-\left\langle z,w\right\rangle |^{n+1+t+c}}\eta (|z|^{2}) dV(z)\leq C\frac{\eta (|w|)^{2}}{(1-|w|^{2})^{c}} \] where \(c>0\) and the estimate is independent of \(w\in B_{n}\). Corresponding estimates are proved in the case \(c<0\) and a logarithmic estimate is established for \(c=0\). Forelli and Rudin proved these estimates in the case \(\eta =1\) by means of power series expansions, orthogonality of monomials and estimates for the gamma function. In the present paper the author gives a very elementary proof by converting to polar coordinates, averaging over the torus, then obtain corresponding elementary integral estimates. For the case \(c>0\) the desired integral estimate takes the form \[ \int_{0}^{1}\frac{(1-r)^{t}}{|1-rx|^{1+t+c}}\eta (r) dr\leq C\frac{\eta (x)}{(1-x)^{c}} \] Clearly the result then relies heavily on the rotation invariance of \(\eta \) . Just as Forelli and Rudin used their estimates to prove \(L^{p}\)-boundedness of Bergman projections, the author proves corresponding bounds for generalized Bergman operators that take into account the Rudin-Forelli weights.