L^{p,q}_ν-boundedness of the Bergman projection on tube domains over symmetric cones

Determine the complete range of parameters p and q (with 1 ≤ p < ∞) and weights ν for which the weighted Bergman projection P_ν on the tube domain T_Ω over an irreducible symmetric cone Ω (of rank r and dimension n) extends to a bounded operator on the mixed-norm space L^{p,q}_ν(T_Ω). Specifically, characterize all (p,q,ν) beyond the currently known range q'_{ν,p} < q < q_{ν,p}, where q_{ν,p} = min{p, p'}·q_ν, q_ν = 1 + ν/(n/r − 1), and 1/p + 1/p' = 1.

Background

The paper studies Bergman-type operators and analytic function spaces in tube domains over symmetric cones. A central operator is the weighted Bergman projection P_ν, the orthogonal projection from L2_ν(T_Ω) onto A2_ν(T_Ω), defined via the Bergman kernel associated with the cone.

For mixed-norm spaces L{p,q}ν(TΩ), only partial boundedness results of P_ν are known in the literature: the projection is bounded for q in the interval (q'{ν,p}, q{ν,p}), with q_{ν,p} determined by p and the cone parameters. Extending boundedness beyond this interval would have significant implications, such as identifying duals of Bergman spaces A{p,q}_ν with A{p',q'}_ν under the canonical pairing.

This problem has attracted attention due to its difficulty and importance for the structure theory of analytic spaces on symmetric cones.

References

The $L_{\nu}{p, q}$ boundedness of the Bergman projection $P_{\nu}$ is still an open problem and has attracted a lot of attention in recent years. Today it is only known that this projection extends to a bounded operator on $L_{\nu}{p, q}$ for general symmetric cones for the range $1 \leq p<\infty, q_{\nu, p}{\prime}<q<q_{\nu, p}, q_{\nu, p}=\min \left{p, p{\prime}\right} q_{\nu}, q_{\nu}=1+\frac{\nu}{n / r-1}$ and $\frac{1}{p}+\frac{1}{p{\prime}=1$ (see , ).

— On Bergman projections and sharp decomposition theorems in tubular and related domains in $C^n$ (2509.22024 - Shamoyan, 26 Sep 2025) in Section “Preliminaries on geometry of tubular domains over symmetric cones, basic lemmas, new analytic spaces” — paragraph “We mention now several known results on Bergman type projections”