We shall discuss optimal conditions on the geometry of the domain responsible for solvability of the Dirichlet problem, or absolute continuity of the harmonic measure with respect to the Lebesgue measure. In rough terms, the question is: do Brownian travelers see pieces of the boundary of the domain according to their Lebesgue size, or, if not, what do they see? In 1916 F. and M. Riesz proved that in a simply connected planar domain rectifiability is sufficient, and over the years this result has been extended to higher dimensions. The centerpoint of our discussion will be a recently proved converse to F. & M. Riesz’ theorem: rectifiability is also necessary for the absolute continuity of the harmonic measure with respect to the Lebesgue measure for n−1 dimensional sets in Rn . We shall also touch upon a more general setting of domains with lower dimensional boundaries.

4pm

Morrey’s inequality in R n is a classical Sobolev embedding that has many important applications, for instance in the regularity theory of elliptic PDEs. Roughly speaking, this inequality asserts that functions in the Sobolev space W1,p(R n ) are H¨older continuous for any n < p < ∞ with an explicit optimal exponent that depends on n and p. In this talk we will present Morrey’s inequality in the more general framework of Dirichlet spaces with (sub-)Gaussian heat kernel estimates. In particular, we will discover that the optimal exponent not only depends on the Hausdorff and the walk dimension, but also on a further invariant of the space. To this end, we will discuss a recent approach to (1, p)- Sobolev spaces via heat semigroups inspired by ideas that go back to work of de Giorgi and Ledoux. If time permits, we will outline some results and conjectures concerning a critical exponent which might be related to other dimensions of interest in the theory of metric measure spaces. This talk is based on joint work with F. Baudoin.