Dirichlet Problems for Harmonic Functions in Half Spaces

Harmonic Bergman Functions on Half-spaces

We study harmonic Bergman functions on the upper half-space of Rn. Among our main results are: The Bergman projection is bounded for the range 1 < p <∞; certain nonorthogonal projections are bounded for the range 1 ≤ p < ∞; the dual space of the Bergman L1-space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range 1 ≤ p <∞; the Bergma...

Half-Dirichlet problems for Dirac operators in Lipschitz domains

Recall that in the case of the Dirichlet problem for the Laplace operator ∂2 x +∂ 2 y in Ω ⊆ R2, one prescribes the whole trace of a harmonic function in, say, L2(∂Ω). On the other hand, for the Cauchy-Riemann operator ∂x + i∂y, natural boundary problems are obtained by prescribing “half” of the trace of the analytic function in L2(∂Ω). Such half-Dirichlet problems arise when, for example, one ...

Invariant Percolation and Harmonic Dirichlet Functions

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of BenjaminiLyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those on the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase f...

Harmonic Polynomials and Dirichlet-Type Problems

We take a new approach to harmonic polynomials via differentiation. Surprisingly powerful results about harmonic functions can be obtained simply by differentiating the function |x|2−n and observing the patterns that emerge. This is one of our main themes and is the route we take to Theorem 1.7, which leads to a new proof of a harmonic decomposition theorem for homogeneous polynomials (Corollar...

Harmonic functions in non-locally convex spaces