Weighted two-parameter Bergman space inequalities

For f, a function defined on R-d1 x R-d2, take a to be its biharmonic extension into R-+(d1+1) x R-+(d2+1). In this paper we prove strong sufficient conditions on measures mu and weights v such that the inequality graphic will hold for all f in a reasonable test class, for 1 < p less than or equal to 2 less than or equal to q < infinity. Our result generalizes earlier work by R, L. Wheeden and the author on one-parameter harmonic extensions. We also obtain sufficient conditions for analogues of (*) to hold when the entries of del(1)del(2)u are replaced by more general convolutions.