Hardy-Type Spaces Arising from a Vector-Valued Cauchy Kernel

Abstract

An integral formula of Cauchy type was recently developed that reproduces any continuous $f:\overline{{\mathbb {B}}} \rightarrow {\mathbb {C}}^n$ that is holomorphic in the open unit ball ${\mathbb {B}}$ of ${\mathbb {C}}^n$ using a fixed vector-valued kernel and the scalar expression $\langle f(u),u \rangle$, where $u\in \partial {\mathbb {B}}$ and $\langle \cdot ,\cdot \rangle$ is the Hermitian inner product in ${\mathbb {C}}^n$, 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.