Expand all Collapse all | Results 1 - 7 of 7 |
1. CMB 2011 (vol 55 pp. 597)
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 |
2. CMB 2011 (vol 55 pp. 242)
Convergence in Capacity In this note we study the convergence of sequences of Monge-AmpÃ¨re measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.
Keywords:complex Monge-AmpÃ¨re operator, convergence in capacity, plurisubharmonic function Categories:32U20, 31C15 |
3. CMB 2004 (vol 47 pp. 481)
A New Characterization of Hardy Martingale Cotype Space We give a new characterization of Hardy martingale cotype
property of complex quasi-Banach space by using the existence of a
kind of plurisubharmonic functions. We also characterize the best
constants of Hardy martingale inequalities with values
in the complex quasi-Banach space.
Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function Categories:46B20, 52A07, 60G44 |
4. CMB 2003 (vol 46 pp. 373)
Potential Theory of the Farthest-Point Distance Function We study the farthest-point 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 |
5. CMB 2003 (vol 46 pp. 268)
Group Cohomology and $L^p$-Cohomology of Finitely Generated Groups Let $G$ be a finitely generated, infinite group, let $p>1$, and let
$L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in
G} |a_x |^p < \infty \}$. In this paper we will study the first
cohomology group of $G$ with coefficients in $L^p(G)$, and the first
reduced $L^p$-cohomology space of $G$. Most of our results will be for a
class of groups that contains all finitely generated, infinite nilpotent
groups.
Keywords:group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functional Categories:43A15, 20F65, 20F18 |
6. CMB 1998 (vol 41 pp. 129)
Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the
weighted Bergman spaces of the unit ball in the $n$-dimensional complex
space is considered and two pluriharmonic symbols of commuting
Toeplitz type operators are completely characterized.
Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators. Categories:47B38, 32A37 |
7. CMB 1997 (vol 40 pp. 60)
Cauchy's problem for harmonic functions with entire data on a sphere We give an elementary potential-theoretic 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 |