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

q

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

condition.

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.