Two New Lipschitz Type Spaces and Their Characterizations

In this paper, we first introduce the weak Lipschitz spaces WLip(q,alpha), 1 < q < infinity, 0 < alpha <1 which are the analog of weak Lebesgue spaces L-q(,infinity) in the setting of Lipschitz space. We obtain the equivalence between the norm parallel to center dot parallel to Lip(alpha) and parallel to center dot parallel to WLip(q,alpha). As an application, we show that the commutator M-beta(b) is bounded from L-p to L-q,L-infinity for some p is an element of (1, infinity) and 1/p - 1/q = alpha+beta/n if and only if b is in Lip(alpha). We also introduce the weak central bounded mean oscillation space WCBMOq,alpha, and give a characterization of WCBMO(q,alpha )via the boundedness of the commutators of Hardy type operators.

Automated summary

In this paper, we introduce the weak Lipschitz spaces WLip(q,alpha), which are analogous to the weak Lebesgue spaces L-q(infinity) in the context of Lipschitz space. We establish the equivalence between the norms parallel to center dot parallel to Lip(alpha) and parallel to center dot parallel to WLip(q,alpha). Moreover, we explore the boundedness of commutators to characterize the weak central bounded mean oscillation space WCBMO(q,alpha).