Research Items (4)
In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on $\mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.
- Oct 2007
Boundedness of the Hardy operator and its adjoint is characterized between Banach function spaces Xq and Lp. By applying a limiting procedure, corresponding boundedness of the geometric mean operator is also derived.
Weight characterization is obtained for the Lp-Xq boundedness of the two dimensional Hardy operator (H2f)(x1,x2)=∫0x1∫0x2f(t1,t2)dt1dt2. By using a limiting procedure as well as by a direct method, the corresponding boundedness of the two dimensional geometric mean operator (G2f)(x1,x2)=exp(1x1x2∫0x1∫0x2lnf(t1,t2)dt1dt2) is obtained. Key Words: Banach function space, Hardy inequality, Hardy operator, geometric mean operator, two dimensional inequality Mathematical Reviews subject classification: 26D10, 26D15