Quaternionic Hardy spaces

Quaternionic Hankel operators and approximation by slice regular functions

“…As in the case of H p spaces of slice regular functions, studied in [11], it can be proven that L p s (∂B) spaces have a natural structure of quaternionic right Banach spaces. In the case p = 2, as in the case of the Hardy space H 2 (B) (see [1] and [2]) the inner product…”

“…For any 0 < p ≤ +∞ a notion of quaternionic H p space is given in [11]. In particular in [11] it is proven that the radial limit of slice regular functions in H p exists on each slice along almost any radius and it belongs to L p (∂B I ) for any 0 < p ≤ +∞. Taking into account that functions in H p (B) satisfy the Representation Formula (4), we easily conclude that their (a.e.)…”

“…In particular, for any 1 ≤ p ≤ +∞, identifying each function with its radial limit, the space H p (B) can be viewed as a subspace of L p (∂B) , in the sequel denoted by H p (∂B). Another fact about functions in H p (B) that we will use in the sequel is that the radial limit of a function that does not vanish identically is almost everywhere nonvanishing (see again [11]). Our next goal is to show that the dual space of slice L 1 functions is the space of slice L ∞ functions.…”

“…where J = (f (e It )) −1 If (e It ) is determined in view of Proposition 2.1 (the fact that f ∈ H 2 (∂B) guarantees that if f ≡ 0, then it is non vanishing almost everywhere at the boundary, see [11]). Using Representation Formula (4), we have By Cauchy-Schwarz inequality we get then…”

“…Here H ∞ (∂B) denotes the space of bounded slice regular functions on B, introduced in [11], viewed as a subspace of L ∞ s (∂B). By compactness, the infimum in equation ( 3) is attained for any ϕ ∈ L ∞ s (∂B): there exists always a regular function g ∈ H ∞ (∂B) such that…”