There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space X equipped with a doubling measure ¯. A generalization of a Sobolev function and its gradient is a pair u 2 L 1 loc (X), 0 g 2 L p (X) such that for every ball B ae X the Poincar'e-type inequality Z B ju \Gamma uB j d¯ Cr `Z oeB g p d¯ ' 1=p holds, where r is the radius of B and oe 1, C ? 0 are fixed constants. Working in the above setting we show that basically...

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

We consider the regularity problem for subelliptic, sphere-valued p-harmonic maps, associated with a system of Hormander vector fields in a bounded domain of IR n . We prove that for p equal to the homogeneous dimension Q, the maps in question are locally Holder continuous. Our method of proof uses an abstract lemma, which serves as a counterpart of the duality of Hardy space and BMO (even though no Hardy spaces are available in this context) and seems to be of independent interest. Its fairly simple proof, bypassing the whole burden of proof of Fefferman's duality theorem, uses just Sobolev inequality, properties of the fundamental solution of the subelliptic Laplace operator, and an abstract version of fractional integration theorem. 1 Introduction The story begins with the paper of Muller, [59], who --- for the sake of an application to nonlinear elasticity --- proved that if the Jacobian determinant J u of a Sobolev map u 2 W 1;n loc (IR n ; IR n ) is nonnegative, then it...

The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.

We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are pre-Sierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for nonnegative L-harmonic functions on these domains and establish upper and lower bounds for the corresponding heat equation.

We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by Saloff-Coste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting of locally compact Dirichlet spaces. We relate these results to what is known as rough path theory by showing that they provide a natural and powerful analytic machinery for construction and study of (random) geometric Hölder rough paths. (In particular, we obtain a simple construction of the Lyons-Stoica stochastic area for a diffusion process with uniformly elliptic generator in divergence form.) Our approach then enables us to establish a number of far-reaching generalizations of classical theorems in diffusion theory including Wong-Zakai approximations, Freidlin-Wentzell sample path large deviations and the Stroock-Varadhan support theorem. The latter was conjectured by T. Lyons in his recent St. Flour lecture. 1

A Hermitian form q on the dual space, g ∗ , of the Lie algebra, g, of a Lie group, G, determines a sub-Laplacian, ∆, on G. It will be shown that Hörmander’s condition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U ′ , is nondegenerate. The subelliptic heat semigroup, e t∆/4, is given by convolution by a C ∞ probability density ρt. When G is complex and u: G → C is a holomorphic function, the collection of derivatives of u at the identity in G gives rise to an element, û(e) ∈ ∗Key words and phrases. Subelliptic, heat kernel, complex groups, universal enveloping algebra, Taylor map.

Saloff-Coste Abstract. We study how bounds on the local geometry of a Riemannian polyhedral complex yield uniform local Poincaré inequalities. These inequalities have a variety of applications, including bounds on the heat kernel and a uniform local Harnack inequality. We additionally consider the example of a complex, X, which has a finitely generated group of isomorphisms, G, such that X/G = Y is a complex consisting of a finite number of polytopes. We show that when this group, G, haspolynomial volume growth, there is a uniform global Poincaré inequality on

Abstract. In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation. 1.

Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.