QK Spaces on the Unit Circle

Abstract

We introduce a new space QK(∂D) of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition on K such that QK(∂D)=BMO(∂D), as well as a general criterion on weight functions K1 and K2, K1≤K2, such that QK1(∂D)QK2(∂D). We also prove that a measurable function belongs to QK(∂D) if and only if it is Möbius bounded in the Sobolev space LK2(∂D). Finally, we obtain a dyadic characterization of functions in QK(∂D) spaces in terms of dyadic arcs on the unit circle.