Littlewood–Paley decompositions related to symmetric cones and Bergman projections in tube domains

Abstract

Starting from a Whitney decomposition of a symmetric cone Ω, analogous to the dyadic partition [2j, 2(j + 1) of the positive real line, in this paper we develop an adapted Littlewood–Paley theory for functions with spectrum in Ω. In particular, we define a natural class of Besov spaces of such functions, Bνp,q, where the role of the usual derivation is now played by the generalized wave operator of the cone normalΔfalse(∂normal∂xfalse). We show that Bνp,q consists precisely of the distributional boundary values of holomorphic functions in the Bergman space Aνp,qfalse(TΩfalse), at least in a ‘good range’ of indices 1 ⩽ q < qν, p. We obtain the sharp qν, p when p ⩽ 2, and conjecture a critical index for p > 2. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors Pν:Lνp,qfalse→Aνp,q, for which our result implies a positive answer when qν, p′ < q < qν, p}. This extends, to general cones, previous work of the authors on the light‐cone. Finally, we focus on light‐cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light‐cones when p is sufficiently large. 2000 Mathematics Subject Classification 42B35, 32M15.

Besov spaces and the boundedness of weighted Bergman projections over symmetric tube domains

“…, ν). In the end, both of these aspects will be seen to rely onto the following remark: the triangular subgroup T in the Iwasawa decomposition G = T K of the structure group G of Ω is sufficient to perform the analysis of many of the results in [2]. That is, in this paper we show that the K part of G can be made to play no role whatsoever.…”

“…We also present a continuous version of these latter spaces which is new even for the case s 1 = · · · = sr considered in [2]. We use these results to discuss multipliers between Besov spaces and the boundedness of the weighted Bergman projection Ps : L p,q s → A p,q s .…”

“…We give a necessary and a sufficient condition on the values of p, q and s for which this identification between A p,q s and B p,q s holds. We also present a continuous version of these latter spaces which is new even for the case s 1 = · · · = sr considered in [2]. We use these results to discuss multipliers between Besov spaces and the boundedness of the weighted Bergman projection Ps : L p,q s…”

“…= V + i Ω over the cone Ω will be defined with respect to the weights ∆ s . The main concern of this paper is the boundedness of the Bergman projection P s : L The goal of this paper is twofold: first, to develop a continuous analogue of the techniques associated with Besov spaces for the cone, presented in [2], and second, to generalize some of the topics covered there in the special case s = (s 1 , . .…”

“…on symmetric tube domains, contained in [2], to the case where the weights are positive powers ∆s . = ∆ adapted to the structure of the cone.…”