Lp-Boundedness of Bergman projections in tube domains over homogeneous cones

Abstract

In this paper, we generalize to all tube domains over homogeneous cones L p -continuity properties of the Bergman projection.

Holomorphic function spaces on homogeneous Siegel domains

“…In this brief section we recall some notions from [89] (see also [64]). Notice that in what follows we shall no longer use the notation of this chapter, unless explicitly stated, in favour of a more conceptual one.…”

“…On the other hand, in e.g. [64], the generalized power functions on Ω are defined by means of the canonical right action of T + on Ω in complete analogy with the case of Ω. We shall follow the latter convention.…”

“…Debertol [38] studied the spaces A p,q s (D) on irreducible symmetric tube domains and investigated their boundary values and the boundedness of the Bergman projectors. Nana and Trojan [64] studied the spaces A p,p s (D), p ∈ [1, ∞[, on irreducible symmetric tube domains, and investigated the boundedness of the Bergman projectors.…”

“…With this convention, the space of boundary values of A p,q s (C + ) embeds canonically in the classical homogeneous Besov spaceḂ −s p,q (R). The second one is that employed in [17,7,21,10,11,64,63,16,13,33]. In this case, the weight ∆ s Ω acts as a multiplier of the measure, and the resulting space A p,q s (D) is then equal to A p,q (s+b/2)/q (D), where b ∈ R r is as above.…”

“…We shall prove some results on the gamma functions on Ω and Ω , most of which are already present in the literature (cf. [46,64]). We shall then introduce our notation for the Riemann-Liouville operators and prove some basic facts about them.…”