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1. CJM 2013 (vol 66 pp. 284)
Random Harmonic Functions in Growth Spaces and Bloch-type Spaces Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces
of harmonic functions in the unit disk and multi-dimensional unit
ball
which admit a two-sided radial majorant $v(r)$.
We consider functions $v $ that fulfill a doubling condition. In the
two-dimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty
(a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$
where $\xi =\{\xi_{ji}\}$ is a sequence of random
subnormal variables and $a_{ji}$ are
real; in higher dimensions we consider series of spherical harmonics.
We will obtain conditions on the coefficients $a_{ji} $ which imply
that $u$ is in $h^\infty_v(\mathbf B)$ almost surely.
Our estimate improves previous results by Bennett, Stegenga and
Timoney, and we prove that the estimate is sharp.
The results for growth spaces can easily be applied to Bloch-type
spaces, and we obtain a similar characterization for these spaces,
which generalizes results by Anderson, Clunie and Pommerenke and by
Guo and Liu.
Keywords:harmonic functions, random series, growth space, Bloch-type space Categories:30B20, 31B05, 30H30, 42B05 |
2. CJM 2013 (vol 66 pp. 641)
Heat Kernels and Green Functions on Metric Measure Spaces We prove that, in a setting of local Dirichlet forms on metric measure
spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent
to the conjunction of the volume doubling propety, the elliptic Harnack
inequality and a certain estimate of the capacity between concentric balls.
The main technical tool is the equivalence between the capacity estimate and
the estimate of a mean exit time in a ball, that uses two-sided estimates of
a Green function in a ball.
Keywords:Dirichlet form, heat kernel, Green function, capacity Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07 |