There are several generalizations of the classical theory of Sobolev spaces as they are
necessary for the applications to Carnot-Caratheodory spaces, subelliptic equations …

A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem
is to describe the relationship between the shape (geo metry) of the drum and its sound (its …

Our main goal in this long survey article is to provide an overview of the theory of complex
fractal dimensions and of the associated geometric or fractal zeta functions, first in the case …

The present research monograph is a testimony to the fact that Fractal Analysis is deeply
connected to numerous areas of contemporary Mathematics. Here, we have in mind, in …

We study the spectral properties of the Laplacian on infinite Sierpiński gaskets. We prove
that the Laplacian with the Neumann boundary condition has pure point spectrum …

We give a criterion for the existence and uniqueness or the non-existence of the diffusions
on a finitely ramified self-similar fractal. In classical examples this criterion is easy to apply …

We consider the analog of the Laplacian on the Sierpinski gasket and related fractals,
constructed by Kigami. A function f is said to belong to the domain of Δ if f is continuous and …

Basic aspects of differential geometry can be extended to various non-classical settings:
Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Wiener …

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous
functions on certain compact metric spaces. The triples are countable sums of triples where …

For a class of fractals that includes the familiar Sierpinski gasket, there is now a theory
involving Laplacians, Dirichlet forms, normal derivatives, Green's functions, and the Gauss …