Polynomial Null Solutions to Bosonic Laplacians, Bosonic Bergman and Hardy Spaces

TL;DR

This paper introduces and analyzes polynomial null solutions to bosonic Laplacians, develops associated Bergman and Hardy spaces, and provides foundational results including orthogonal decompositions, reproducing kernels, and growth estimates.

Contribution

It is the first to define bosonic Bergman and Hardy spaces in higher spin contexts and to establish their fundamental properties and connections to classical harmonic analysis.

Findings

Orthogonal decomposition of polynomial spaces via null solutions

Reproducing kernels for bosonic Bergman spaces in the unit ball

Growth estimates and measure space connections for bosonic Hardy spaces

Abstract

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.