QK Spaces on the Unit Circle. (20th August 2014) - Zhou, Jizhen
Abstract : We introduce a new spaceQ K ( ∂ D ) of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition onK such thatQ K ( ∂ D ) = BMO ( ∂ D ), as well as a general criterion on weight functionsK 1 andK 2, K 1 ≤ K 2, such thatQ K 1 ( ∂ D )  Q K 2 ( ∂ D ) . We also prove that a measurable function belongs toQ K ( ∂ D ) if and only if it is Möbius bounded in the Sobolev spaceL K 2 ( ∂ D ) . Finally, we obtain a dyadic characterization of functions inQ K ( ∂ D ) spaces in terms of dyadic arcs on the unit circle.