Ap -Inner Functions
In this chapter, we introduce the notion of A p α -inner functions and prove a growth estimate for them. The A p α -inner functions are analogous to the classical inner functions which play an important role in the factorization theory of the Hardy spaces. Each A p α -inner function is extremal for a z-invariant subspace, and the ones that arise from subspaces given by finitely many zeros are called finite zero extremal functions (for α = 0, they are also called finite zero-divisors). In the unweighted case α = 0, we will prove the expansive multiplier property of A up -inner functions, and obtain an “inner-outer”-type factorization of functions in A p . In the process, we find that all singly generated invariant subspaces are generated by its extremal function. In the special case of p = 2 and α = 0, we find an analogue of the classical Carathéodory-Schur theorem: the closure of the finite zero-divisors in the topology of uniform convergence on compact subsets are the A2-subinner functions. In particular, all A2-inner functions are norm approximable by finite zero-divisors.
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