The orthogonal projection on slice functions on the quaternionic sphere

Abstract

Abstract. We study the L p norm of the orthogonal projection from the space of quaternion valued L 2 functions to the closed subspace of slice L 2 functions.The aim of this short note is to study the orthogonal projection Π from the space of quaternion valued L 2 functions to its closed subspace of slice L 2 functions. In particular, to compute the norm of the projection operator we will first show that we can write Π in terms of a quaternionic slice Poisson kernel.Let H = R + Ri + Rj + Rk denote the non commutative 4-dimensional real algebra of quaternions and let B = {q ∈ H : |q| < 1} be the unit ball in H. Its boundary ∂B contains elements of the form q = e It , I ∈ S, t ∈ R, where S = {q ∈ H : q 2 = −1} is the two dimensional sphere of imaginary units in H. We endow ∂B with the measure dΣ e It = dσ(I)dt, which is naturally associated with the Hardy space H 2 (B) of the slice regular functions on B, see [1]. We here normalize the measures so to have Σ(∂B) = σ(S) = 1. The functions we are concerned with here satisfy algebraic conditions and size conditions. A function f : ∂B → H is a slice function if for any I, J ∈ SThat is, slice functions are affine in the S variable:where a, b : ∂B → H are functions depending on t alone. Condition (2) has an algebraic nature. Equation (1) also provides a formula to extend a function f I defined on a slice ∂B I = ∂B∩(R+RI) to a slice function f := ext(f I ) defined on the entire sphere ∂B. Slice functions were introduced in the more general setting of real alternative algebras by Ghiloni and Perotti in [5]. A notable class of slice functions is provided by the restriction to the unit sphere in H of slice regular functions. Such functions have been object of intensive research

Entire Slice Regular Functions

“…Several function spaces have been studied in this framework. In particular, the quaternionic Hardy spaces H 2 (Ω), where Ω is the quaternionic unit ball B or the half space H + of quaternions with positive real part, together with the Blaschke products are in [14,15,16] and further properties are in [28,29,30]. The Hardy spaces H p (B), p > 2, are considered in [144].…”

Quaternionic Hankel operators and approximation by slice regular functions

“…In this paper we will not investigate the general theory of these function spaces, in particular we will focus our attention to their subspaces of slice functions. We recall here that in [7] the L p norm of the orthogonal projection from the space L 2 (∂B) to its closed subspace L 2 s (∂B) is studied. A first (easy) remark is that the restriction to a single slice of an element of L p (∂B) does not necessarily belong to L p (∂B I ) for every I ∈ S (but only for σ-almost every I), while clearly the opposite implication holds true.…”

Slice regular Besov spaces of hyperholomorphic functions and composition operators

“…Several function spaces of the slice hyperholomorphic functions are studied. The quaternionic Hardy spaces are studied in [6,7,8,12,13,14,51]. The Bergman spaces of slice hyperholomorphic functions are invesigated in [19,20,21].…”