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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 Hardys 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.
In this note we continue giving the characterisation 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 inequalitie
We consider a general class of metric measure spaces equipped with a regular Dirichlet form and then provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a generalization of th
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet
In this paper, we will prove the Weyls law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincare-Sobolev, Hardy-Sobolev-Mazya, Adams and Hardy-Adams inequalities on real hyperbolic