Bergman and Poly-Bergman Spaces

Abstract

We start by recalling an old and well-known result. Let $H_ + ^2 \left( \mathbb{R} \right)\left( { \subset L_2 \left( \mathbb{R} \right)} \right)$ be the Hardy space on the upper half-plane II in ℂ, which by definition consists of all functions ϕ on ℝ admitting analytic continuation in II and satisfying the condition

$$ \mathop {\sup }\limits_{v > 0} \int_{\mathbb{R} + iv} {\left| {\phi \left( {u + iv} \right)} \right|^2 du < \infty .} $$

Let $P_\mathbb{R}^ +$ be the (orthogonal) Szegö projection of L₂(ℝ) onto $H_ + ^2 \left( \mathbb{R} \right)$. Then: the Fourier transform F gives an isometric isomorphism of the space L ₂ (ℝ), under which

• 1. the Hardy space$H_ + ^2 \left( \mathbb{R} \right)$ is mapped onto L₂(ℝ₊),

  $$ F:H_ + ^2 \left( \mathbb{R} \right) \to L_2 \left( {\mathbb{R}_ + } \right), $$

• 2. the Szegö projection $P_\mathbb{R}^ +$: L₂(ℝ)→$H_ + ^2 \left( \mathbb{R} \right)$ is unitary equivalent to the projection

  $$ F:P_\mathbb{R}^ + F^{ - 1} - \chi + I. $$