Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1

We analyze the spectrum of a self-adjoint operator on a Laakso space using the projective limit construction originally given by Barlow and Evans. We will use the hierarchical cell structure induced by the choice of approximating quantum graphs to calculate the spectrum with multiplicities. We also extend the method for using the hierarchical cell structure to more general projective limits beyond Laakso spaces. MCS: 34L40 (primary); 34L16; 54B30 1

The purpose of these notes is to prove the classical Rademacher’s Theorem. There are versions that are far more general, notably Cheeger obtained recently a Rademacher type theorem only assuming the space has a Vitali Covering Theorem, which incidentally gives a new proof of the classical

Abstract. For a metric space X, we study the space D ∞ (X) of bounded functions on X whose infinitesimal Lipschitz constant is uniformly bounded. D ∞ (X) is compared with the space LIP ∞ (X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D ∞ (X) with the Newtonian-Sobolev space N 1, ∞ (X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D ∞ (X) = N 1, ∞ (X). 1.

Let E be a closed set in R n, and suppose that there is a k ≥ 1 such that every x, y ∈ E can be connected by a rectifiable path in E with length ≤ k |x−y|. This condition is satisfied by chord-arc curves, Lipschitz manifolds of any dimension, and fractals like Sierpinski gaskets and carpets. Note that length-minimizing paths in E are chord-arc curves with constant k. A basic feature of this condition is that one can integrate local Lipschitz conditions on E to get global conditions. For instance, if f: E → R is locally Lipschitz of order 1 with constant C ≥ 0, then f is globally Lipschitz on E with constant k C. Let A(x) be a continuous function on E with values in linear mappings from Rn to R. Equivalently, one can use the standard inner product on Rn to represent A(x) by a continuous mapping from E into Rn. Also let f be a locally Lipschitz real-valued function on E. Suppose that A includes the directional derivatives of f almost everywhere on any rectifiable curve in E, in the sense that the derivative of f(p(t)) is equal to A(p(t)) applied to the derivative of p(t) for almost every t when p(t) is a

Very often in analysis, one focuses on connected spaces. This is certainly not always the case, and in particular there are many interesting matters related to Cantor sets. Here we are more concerned with a type of complementary situation. As a basic scenario, suppose that U is an open set in R n and that E is a closed set contained in the boundary of U such that for every x∈U and r> 0 there is a connected component of U contained in the ball with center x and radius r. For instance, E might be the boundary of U. As a uniform version of the condition, one might ask that there be a constant C> 0 such that the aforementioned connected component of U contains a ball of radius C −1 r. The connected components of U might be quite regular, even if there are infinitely many of them. As a basic example, E could be a Cantor set in the real line, and U could be the complement of E or the union of the bounded complementary components of E. This does not work in higher dimensions, where the complement of a Cantor set is connected. Sierpinski gaskets and carpets in the plane are

In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincaré inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger’s work these upper gradients can be used to define a Sobolev space. We show that this leads to a Dirichlet form. The second construction was introduced by Barlow and Evans in 2004 as a way of producing exotic spaces along with Markov processes from simpler spaces and processes. We show that for the correct base process in the Barlow Evans construction that this Markov process corresponds to the Dirichlet form derived from the minimal generalized upper gradients. MSC Codes: 31C25 (Primary) 60J45, 28A80, 46A13 1