Abstract. In this paper we establish improved Hardy and Rellich type inequalities on a Riemannian manifold M . Furthermore, we also obtain sharp constants for improved Hardy and Rellich type inequalities on the hyperbolic space H n .

We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities on a Riemannian manifold M , started in [17]. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as improved version of our Uncertainty principle inequalities on a Riemannian manifold M . In particular, we obtain sharp constants for these inequalities on the hyperbolic space H n .

Abstract. In this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation:where ∆ G,p is the p-sub-Laplacian on Carnot group G and V ∈ L 1 loc (Ω).

We find a simple sufficient criterion on a pair of nonnegative weight functions V (x) and W (x) on a Carnot group G, so that the general weighted L p Hardy type inequality G V (x) |∇ G φ (x)| p dx ≥ G W (x) |φ (x)| p dx is valid for any φ ∈ C ∞ 0 (G) and p > 1. It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on G. We also present some new results on two-weight L p Hardy type inequalities with remainder terms on a bounded domain Ω in G via a differential inequality.

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Key wordsIn this paper, we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equations:Here, Ω is a Carnot-Carathéodory metric ball in R N and V ∈ L 1 loc (Ω). The critical exponents m * and p * are found, and the nonexistence results are proved for m * ≤ m < 1 and p