We prove by methods of harmonic analysis a result on the existence of solutions for twisted cohomological equations on translation surfaces with loss of derivatives at most $3+$ in Sobolev spaces. As a consequence we prove that product translation flows on (three-dimensional) translation manifolds which are products of a (higher-genus) translation surface with a (flat) circle are stable in the sense of A. Katok. In turn, our result on product flows implies a stability result of time- $\tau $ maps of translation flows on translation surfaces.

If $f$ is a real-valued function on $\left[ -\pi ,\,\pi \right]$ that is Henstock-Kurzweil integrable, let ${{u}_{r}}(\theta )$ be its Poisson integral. It is shown that ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,o\left( 1/\left( 1-r \right) \right)$ as $r\,\to \,1$ and this estimate is sharp for $1\,\le \,p\,\le \,\infty $. If $\mu $ is a finite Borel measure and ${{u}_{r}}(\theta )$ is its Poisson integral then for each $1\,\le \,p\,\le \,\infty $ the estimate ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,O\left( {{\left( 1-r \right)}^{1/p-1}} \right)$ as $r\,\to \,1$ is sharp. The Alexiewicz norm estimates $\left\| {{u}_{r}} \right\|\,\le \,\left\| f \right\|$$\left( 0\,\le \,r\,<\,1 \right)$ and $\left\| {{u}_{r}}-f \right\|\,\to 0\left( r\to 1 \right)$ hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock-Kurzweil integrable boundary data. There are similar growth estimates when $u$ is in the harmonic Hardy space associated with the Alexiewicz norm and when $f$ is of bounded variation.

Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

We introduce the class of Harnack domains in which a Harnack type inequality holds for positive harmonic functions with bounds given in terms of the distance to the domain's boundary. We give conditions connecting Harnack domains with several different complete metrics. We characterize the simply connected plane domains which are Harnack and discuss associated topics. We extend classical results to Harnack domains and give applications concerning the rate of growth of various functions defined in Harnack domains. We present a perhaps new characterization for quasidisks.

Given a domain D in R” and two specified points P0 and P1 in D we consider the problem of minimizing u(p1) over all functions harmonic in D with values between 0 and 1 normalised by the requirement u(P0) = 1/2. We show that when D is suitably regular the problem has a unique solution u* which necessarily takes on boundary values 0 or 1 almost everywhere on the boundary. In the process we prove that it is possible to separate P0 and P1by a harmonic function whose boundary value is supported in an arbitrary set of positive measure. These results depend on the fact that (under suitable regularity conditions) a harmonic function which vanishes on an open subset of the boundary has a normal derivative which is almost everywhere non-vanishing in that set.

The Poisson integral of a Denjoy-Perron integrable function defined on the boundary of an open disc is harmonic in this disc. Moreover, almost everywhere on the boundary, the nontangential limits of the integral coincide with the boundary condition. This extends the classical result for Lebesgue integrable boundary conditions. By means of conformai maps, a generalization to domains bounded by a sufficiently smooth Jordan curve is also obtained.