Existence of bounded Bergman projection T_β on mixed-norm spaces over products of tube domains

Establish whether, for products of tube domains T_Ω^m over an irreducible symmetric cone Ω, there exists a sufficiently large parameter β such that the Bergman-type projection T_β is bounded from the mixed-norm measurable space L^{p,q}_α(T_Ω^m) to its analytic subspace A^{p,q}_α(T_Ω^m) for all p, q ≥ 1, where the norms are defined by integrating the p-th power over V and then the q-th power over Ω with weights Δ^{β_j}(y_j) and β_j > −1 for j = 1,…,m.

Background

The authors introduce and discuss mixed-norm analytic spaces A^{p,q}_α on products of tube domains T_Ω^m, defined by iterated integration with determinant weights Δ^{β_j}(y_j). They also consider the corresponding larger measurable classes L^{p,q}_α obtained by replacing analytic functions with measurable ones.

While boundedness results for certain Bergman-type operators are proved in single tube domains and hinted at for products, a general bounded projection T_β from L^{p,q}α(TΩ^m) to A^{p,q}α(TΩ^m) remains unresolved. The question is whether choosing β sufficiently large yields a bounded operator across all admissible p, q ≥ 1 and weights β_j > −1.

Resolving this would extend Bergman projection theory to mixed-norm settings on product domains and unify several operator-theoretic results across complex multivariable function spaces.