Weighted two-parameter Bergman space inequalities

Abstract

For f , a function defined on Rd1 ×Rd2 , take u to be its biharmonic extension into R+ +1 × Rd2 +1 . In this paper we prove strong d1 + sufficient conditions on measures µ and weights v such that the inequality 1/q q ∇2 u dµ(x1 , x2 , y1 , y2 ) d +1 d +1 R+1 ×R+2 1/p ≤ f p v dx Rd1 ×Rd2 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 ∇2 u are replaced by more general convolutions.