This paper is a study
of bounded point derivations on the classical Banach algebras of analytic
functions of a complex variable. The results are positive in character. The
higher-order Gleason metrics dp of R(X) are introduced and conditions are
studied under which convergence takes place with respect to these metrics.
In particular, if R(X) admits a pth-order bounded point derivation at a
point x ∈ ∂X and X satisfies a cone condition at x, then dp(y,x) tends to
0 as y tends to x along the midline of the cone. Similar results hold for
the other classical function algebras. In the case of the algebra H∞(U),
for open U ⊂ C, the analogous results hold only for regular derivations (a
regular p-th-order derivation maps zp to a nonzero complex number). The
points of the maximal ideal space of H∞(U) at which regular bounded point
derivations exist are characterized in terms of analytic capacity, following
Hallstrom.
