10 BAERNSTEIN AND SAWYER
This is the H (R ) version of Zygmund's L log L theorem on the
circle. We are surprised that no one seems to have discovered it before.
It includes the Taibleson-Weiss theorem for p = 1 , since K c Y for
ql and an(l - —) . The examples f. . in §5 show that log |x|
cannot be replaced by any essentially smaller weight,, and the examples f
show that L log L cannot be replaced by any essentially weaker integrability
Theorems la,b are about functions with possible non-integrable
singularity at the origin. By translation invariance., the conclusions still
hold if the rings A, are replaced by rings 2 |x-x | 2 .I t
would be interesting to find embedding theorems in which the one point
singular set jx } is replaced by a substantially larger one.