Polar and AC operators, the Hilbert transform, and matrix-weighted shifts
Julie Wilson · ERA · 1997
Well-bounded operators of type (B) are the building blocks for trigonometrically wellbounded, polar and AC operators.We examine the relationship between polar and AC operators of type (B) and explore the concepts of bounded variation and absolute continuity for functions defined on annuli.In 1973, Hunt, Muckenhoupt and Wheeden showed that the Hubert transform is a bounded operator on a weighted LP space precisely when the weight satisfies the A condition.This result is proved independently for LP spaces over the reals, the circle and the integers.We investigate the inter-relationships between these three theorems and show that the theorems for the reals and the integers are equivalent.When W satisfies the A condition on It, as well as certain boundedness conditions, we show that the one-parameter group of translation operators {U i } €a on LP (R) defined by Utf(s) = f(s + t) admits a Stone-type integral representation -that is Ut = limWe show that for fixed t the operator Ut is bounded on L(R) if and only if