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We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space. [ABSTRACT FROM AUTHOR]

We present certain Liouville properties of eigenfunctions of secondorder elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of secondorder elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schrödinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), pp. 75-84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture. [ABSTRACT FROM AUTHOR]