Resumen de Weighted two-parameter Bergman space inequalities

J. Michael Wilson

• For f, a function defined on Rd1 ×Rd2 , take u to be its biharmonic extension into Rd1+1 + × Rd2+1 + . In this paper we prove strong sufficient conditions on measures µ and weights v such that the inequality (∗) Rd1+1 + ×Rd2+1 + |∇1∇2u| q dµ(x1, x2, y1, y2) 1/q ≤  Rd1 ×Rd2 |f| pv dx1/p will hold for all f in a reasonable test class, for 1 < p ≤ 2 ≤ q < ∞. 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 ∇1∇2u are replaced by more general convolutions.