Hardy-Type Spaces Arising from a Vector-Valued Cauchy Kernel
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Hardy-Type Spaces Arising from a Vector-Valued Cauchy Kernel Jerry R. Muir Jr. 1
Received: 5 December 2016 / Accepted: 7 April 2017 © Springer-Verlag Berlin Heidelberg 2017
Abstract An integral formula of Cauchy type was recently developed that reproduces any continuous f : B → Cn that is holomorphic in the open unit ball B of Cn using a fixed vector-valued kernel and the scalar expression f (u), u, where u ∈ ∂B and ·, · is the Hermitian inner product in Cn , which is key to defining the numerical range of f . We consider Hardy-type spaces associated with this vector-valued kernel. In particular, we introduce spaces of vector-valued holomorphic mappings properly containing the vector-valued Hardy spaces that are reproduced through the process described above and isomorphic spaces of scalar-valued non-holomorphic functions that satisfy many of the familiar properties of Hardy space functions. In the spirit of providing a straightforward introduction to these spaces, proof techniques have been kept as elementary as possible. In particular, the theory of maximal functions and singular integrals is avoided. Keywords Hardy spaces · Cauchy–Szegö kernel · Reproducing kernels · Holomorphic mappings Mathematics Subject Classification Primary 32A35 · 32A26; Secondary 46E15 · 46E22 · 46E40 · 32H02
Communicated by Filippo Bracci.
B 1
Jerry R. Muir Jr. [email protected] Department of Mathematics, The University of Scranton, Scranton, PA 18510, USA
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J. R. Muir Jr.
1 Introduction The well-known Cauchy–Szegö kernel for the open unit ball B ⊆ Cn is the function C : B × S → C given by C(z, u) =
1 , (1 − z, u)n
(1.1)
where S = ∂B is the unit sphere and ·, · is the Hermitian inner product in Cn . If A(B) denotes the ball algebra, consisting of those continuous f : B → C that are holomorphic in B, and σ is the unique rotation-invariant Borel probability measure on S (i.e., normalized Lebesgue surface area measure), then the Cauchy integral formula on B is (1.2) f (z) = C(z, u) f (u) dσ (u), z ∈ B, f ∈ A(B). S
(See [8,12].) Cauchy integrals provide the key to one of the two typical ways of defining the Hardy spaces on B. Specifically, we may define H p (B) for p ≥ 1 as the space of those holomorphic f : B → C such that sup
0 1. Even though α>1 Dα (u) = B, K-lim f (u) may exist even when limB z→u f (z) does not. It is known [8,12] that if f ∈ H p (B), p ≥ 1, then f has a finite K-limit at almost every u ∈ S. Letting f ∗ (u) = K-lim f (u) for these u, f ∗ ∈ L p (σ ) and
f ∗ p = f H p (B) . That is, H p (B) can be isometrically embedded as a (closed) subspace of L p (σ ). For this reason, we will now write · p for the norm in H p (B). (Note that we use the notation f ∗ (u) for the radial limit and for the stronger K-limit; it will be clear which limit is intended whenever it matters.) Functions f ∈ H p (B) are reproduced by f = C[ f ∗ ],
(2.2)
where C is the Cauchy transform, defined on L 1 (σ ) by C[g](z) =
S
C(z, u)g(u) dσ (u), z ∈ B.
(2.3)
When p = 2, C is the orthogonal (Szegö) projection o
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