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1. CJM 2006 (vol 58 pp. 691)
Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups On a compact connected group $G$, consider the infinitesimal
generator $-L$ of a central symmetric Gaussian convolution
semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution
and smooth function spaces, we prove that $L$ is hypoelliptic if and only if
$(\mu_t)_{t>0} $ is absolutely continuous with respect to Haar measure
and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that
$\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds
if and only if any Borel measure $u$ which is solution of $Lu=0$
in an open set $\Omega$ can be represented by a continuous
function in $\Omega$. Examples are discussed.
Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60 |
2. CJM 2005 (vol 57 pp. 99)
Second Order Operators on a Compact Lie Group We describe the structure of the space of second order elliptic
differential operators on a homogenous bundle over a compact Lie
group. Subject to a technical condition, these operators are
homotopic to the Laplacian. The technical condition is further
investigated, with examples given where it holds and others where
it does not. Since many spectral invariants are also homotopy
invariants, these results provide information about the invariants
of these operators.
Categories:58J70, 43A77 |
3. CJM 2004 (vol 56 pp. 431)
Group Actions and Singular Martingales II, The Recognition Problem We continue our investigation in [RST] of a martingale formed by picking a
measurable set $A$ in a compact group $G$, taking random rotates of $A$, and
considering measures of the resulting intersections, suitably normalized. Here
we concentrate on the inverse problem of recognizing $A$ from a small amount of
data from this martingale. This leads to problems in harmonic analysis on $G$,
including an analysis of integrals of products of Gegenbauer polynomials.
Categories:43A77, 60B15, 60G42, 42C10 |