Weighted inequalities for Hardy-type operators and their application in the Interplation Theory

We study real interpolation spaces (Xo, X1) 12,q, where {} is a parameter function, not necessarily a power weight. Using a discretization method we "discretize" the norm in (Xo, X1) 12,q. The resulting norm is given by the corresponding quasiconcave function h and its discretizing sequence, we denote the space endowed with this norm by (Xo, X1)h,q· We give a direct proof of a theorem dueto V. I. Ovchinnikov and A. S. Titenkov, which characterizes the space (Lp0 , Lp1 )h,q in terms of the non- increasing rearrangement. Further, we find a relation between the dilation indices of a quasiconcave function h and its discretizing sequence. In the case when the dilation indices of h are not limiting, the space ( Lp0 , Lp1 ) h,q coincides wi th some classical Lorentz space A q ( r.p). If the dilation indices are limiting, then we characterize the space (Lp0 , Lp1 )h,q as an extrapolation space. Powered by TCPDF (www.tcpdf.org