1. CMB 2016 (vol 60 pp. 146)
 Khavinson, Dmitry; Lundberg, Erik; Render, Hermann

The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions
It is shown that the Dirichlet problem for the slab $(a,b) \times
\mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof
is based
on a generalized Schwarz reflection principle. Moreover, it is
shown that
for a given entire harmonic function $g$
the inhomogeneous difference equation $h
( t+1,y) h (t,y) =g ( t,y)$
has an entire harmonic solution $h$.
Keywords:reflection principle, entire harmonic function, analytic continuation Categories:31B20, 31B05 

2. CMB 2015 (vol 59 pp. 211)
 Totik, Vilmos

Universality Under SzegÅ's Condition
This paper presents a
theorem on universality on orthogonal polynomials/random matrices
under a weak local condition on the weight function $w$.
With a new inequality for
polynomials and with the use of fast decreasing polynomials,
it is shown that an approach of
D. S. Lubinsky is applicable. The proof works
at all points which are Lebesguepoints both
for the weight function $w$ and for $\log w$.
Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory Categories:42C05, 60B20, 30C85, 31A15 

3. CMB 2012 (vol 56 pp. 241)
4. CMB 2011 (vol 55 pp. 597)
 Osękowski, Adam

Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$,
$1 < p < \infty$, for which the following holds. If $u$, $v$ are
orthogonal harmonic functions on a Euclidean domain such that $v$ is
differentially subordinate to $u$, then
$$ \v\_p \leq C_{p,\infty}
\u\_\infty,\quad
\v\_1 \leq C_{1,p} \u\_p.
$$
In particular, the inequalities are still sharp for the conjugate
harmonic functions on the unit disc of $\mathbb R^2$.
Sharp probabilistic versions of these estimates are also studied.
As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.
Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem Categories:31B05, 60G44, 60G40 

5. CMB 2011 (vol 55 pp. 242)
 Cegrell, Urban

Convergence in Capacity
In this note we study the convergence of sequences of MongeAmpÃ¨re measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.
Keywords:complex MongeAmpÃ¨re operator, convergence in capacity, plurisubharmonic function Categories:32U20, 31C15 

6. CMB 2009 (vol 52 pp. 555)
 Hirata, Kentaro

Boundary Behavior of Solutions of the Helmholtz Equation
This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewoodtype theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
Keywords:boundary behavior, Helmholtz equation Categories:31B25, 35J05 

7. CMB 2009 (vol 52 pp. 105)
8. CMB 2008 (vol 51 pp. 229)
 Hanley, Mary

Existence of Solutions to Poisson's Equation
Let $\Omega$ be a domain in $\mathbb R^n$ ($n\geq 2$). We find a
necessary and sufficient topological condition on $\Omega$ such
that, for any measure $\mu$ on $\mathbb R^n$, there is a function $u$
with specified boundary conditions that satisfies the Poisson
equation $\Delta u=\mu$ on $\Omega$ in the sense of distributions.
Category:31B25 

9. CMB 2005 (vol 48 pp. 133)
 Talvila, Erik

Estimates of HenstockKurzweil Poisson Integrals
If $f$ is a realvalued function on $[\pi,\pi]$ that
is HenstockKurzweil integrable, let $u_r(\theta)$ be its Poisson
integral. It is shown that $\u_r\_p=o(1/(1r))$ as $r\to 1$
and this estimate is sharp for $1\leq p\leq\infty$.
If $\mu$ is a finite Borel measure and $u_r(\theta)$ is its Poisson
integral then for each $1\leq p\leq \infty$ the estimate
$\u_r\_p=O((1r)^{1/p1})$ as $r\to 1$ is sharp.
The Alexiewicz
norm estimates $\u_r\\leq\f\$ ($0\leq r<1$) and $\u_rf\\to 0$
($r\to 1$) hold. These estimates lead to two uniqueness theorems for
the Dirichlet problem
in the unit disc with HenstockKurzweil 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.
Categories:26A39, 31A20 

10. CMB 2003 (vol 46 pp. 373)
 Laugesen, Richard S.; Pritsker, Igor E.

Potential Theory of the FarthestPoint Distance Function
We study the farthestpoint distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width Categories:31A05, 52A10, 52A40 

11. CMB 2003 (vol 46 pp. 252)
12. CMB 2002 (vol 45 pp. 154)
 Weitsman, Allen

On the Poisson Integral of Step Functions and Minimal Surfaces
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.
Keywords:harmonic mappings, dilatation, minimal surfaces Categories:30C62, 31A05, 31A20, 49Q05 

13. CMB 1998 (vol 41 pp. 257)
14. CMB 1997 (vol 40 pp. 60)
 Khavinson, Dmitry

Cauchy's problem for harmonic functions with entire data on a sphere
We give an elementary potentialtheoretic proof of a theorem of
G.~Johnsson: all solutions of Cauchy's problems for the Laplace
equations with an entire data on a sphere extend harmonically to
the whole space ${\bf R}^N$ except, perhaps, for the center of the
sphere.
Keywords:harmonic functions, Cauchy's problem, homogeneous harmonics Categories:35B60, 31B20 
