Three Related Problems of Bergman Spaces of Tube Domains over Symmetric Cones

International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 10, Issue 2, pp: (1-13), Month: October 2022 - March 2023, Available at: www.researchpublish.com

Three Related Problems of Bergman Spaces of Tube Domains over Symmetric Cones Sami Ali (1), Obaid.B. A. A (2), Ahmed Sufyan Abakar (3), Shawgy Hussein (4) (1) 2)

University of Al-Butana, Faculty of education, Department of Physical and Mathematics,

Sudan University of Science and Technology, College of Science, Department of Mathematics, Sudan 3)

4)

Ministry of education Sultante of Oman

Sudan University of Science and Technology, College of Science, Department of Mathematics, Sudan DOI: https://doi.org/10.5281/zenodo.7157247

Published Date: 07-October-2022

Abstract: The Szego projection of tube domains over irreducible symmetric cones is unbounded in 𝑳(𝟏+𝝐) . Indeed, this is a consequence of the fact that the characteristic function of a disc is not a Fourier multiplier, a fundamental theorem proved by C. Fefferman in the 70's. The same problem, related to the Bergman projection, deserves a different approach. In this survey, based on joint work of the author with D. Bekolle, G. Garrigos, M. Peloso and F. Ricci, we give partial results on the range of 𝟏 + 𝝐 for which it is bounded. We also show that there are two equivalent problems, of independent interest. One is a generalization of Hardy inequality for holomorphic functions. The other one is the characterization of the boundary values of functions in the Bergman spaces in terms of an adapted Littlewood–Paley theory. This last point of view leads naturally to extend the study to spaces with mixed norm as well. Keywords: Whitney decomposition; Symmetric cone; Bergman projector; Littlewood – Paley; Hardy inequality.

1. INTRODUCTION For 𝑉 be an irreducible symmetric cone in the Euclidean space V , and 𝑇Ω = 𝑉 + 𝑖𝛺the corresponding tube domain in the complexified space𝑉 ℂ . We shall note 𝑛 the dimension of 𝑉 and 𝑟 the rank of Ω . Moreover , we shall denote by (𝑥|𝑦) the scalar product inV, and by Δ the determinant function . For the description of such cones in terms of Jordan , one may use the book of Faraut and Koranyi [8] . One may also have in mind the typical example that one obtains when 𝑉 is the space of real symmetric 𝑟 × 𝑟 matrices and Ω is the cone of positive definite matrices . In this example , the scalar product on 𝑉 is induced by the Hilbert-Schmidt norm of the matrices , and the determinant function is given by the determinant of the matrices . The rank is 𝑟 , while the dimension is

𝑟(𝑟+1) 2

.

We shall also make use of the generalized wave operator on 𝑉, given by ∎ = Δ(

1 𝜕 ) 𝑖 𝜕𝑥

This is a differential operator of degree 𝑟 , defined by the equality 1 𝜕 ( ) [∑ 𝑒 𝑖(𝑥⁄𝜉𝑘 ) ] = ∑ Δ(𝜉𝑘 )𝑒 𝑖(𝑥⁄𝜉𝑘 ) , 𝜉𝑘 ∈ 𝑉 𝑖 𝜕𝑥 𝑘 𝑘 It is the usual derivative (up to a constant) when 𝛺 is the half-line (0, ∞). Its name is due to another fundamental example, given by the forward light cone in 𝑅𝑛 ,

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