| Let f be an analytic function in the unit ball B of (//C)('N) for which the "mixed norm";is finite. Here 0 < p < (INFIN), 0 < q < (INFIN), (omega) is a suitable radial weight func- tion, and (sigma) is normalized Lebesgue measure on (PAR-DIFF)B. Note that when p = q, because of the "polar coordinates" formula, the space of all such functions is just the Bergman space with weight (omega). General mixed norm spaces were studied extensively by Benedek and Panzone.;We begin by generalizing a collection of results gotten by Luecking for the Bergman spaces or the Hardy spaces. Bounded- ness of certain Bergman projections is proven first, using vector-valued integration and some facts due to Forelli and Rudin. Repre- sentation of the dual space of our mixed norm spaces follows from this. Then a representation of functions in our mixed norm spaces is obtained (by using duality) and several equivalent norms are produced (by refining arguments of Luecking).;We also state a general "Carleson measure theorem" for our mixed norm spaces whose proof depends largely on geometry and the connection between the nonisotropic metric and the invariant Poisson kernel. Several consequences are noted, including a theo- rem originally due to Cima and Wogen. Other related methods and results are given, among them a Carleson measure theorem for mixed norm spaces in the polydisc, a generalization of a result of Hastings.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)... |
