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Some gradient estimates on covering manifolds

Dungey N.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

437-443

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.

Generalized sturm separation theorem

Przeworska-Rolewicz D.

Demonstratio Mathematica

2003

Vol. 36, nr 3

735--746

True shifts for right invertible operators has been examined in several papers in various aspects (cf. PR[4], PR[5]). A generalization of Sturm separation theorem was given in PR[2] in the case when a right invertible operator under consideration had the one-dimensional kernel. Following the preprint [6], it is shown that the Sturm theorem holds without any assumption about the dimension of that kernel. In the last section of the present paper there are considered the multiplicative symbols in Leibniz algebras.