QK spaces on the unit circle

We introduce a new space qk(∂double-struck 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(∂double-struck d) = bmo(∂double-struck d),as well as a general criterion on weight functions k1 and k2,k1≤ k2,such that qk1(∂double-struck d) subset of with not equal to qk2(∂double-struck d). we also prove that a measurable function belongs to qk(∂double-struck d) if and only if it is möbius bounded in the sobolev space lk 2(∂double-struck d). finally,we obtain a dyadic characterization of functions in qk(∂double-struck d) spaces in terms of dyadic arcs on the unit circle. copyright © 2014 jizhen zhou.