1. CMB Online first
 Steinerberger, Stefan

An Endpoint Alexandrov Bakelman Pucci estimate in the Plane
The classical AlexandrovBakelmanPucci estimate
for the Laplacian states
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c_{s,n} \operatorname{diam}(\Omega)^{2\frac{n}{s}}
\left\ \Delta u
\right\_{L^s(\Omega)}$$
where $\Omega \subset \mathbb{R}^n$, $u \in C^2(\Omega) \cap
C(\overline{\Omega})$ and $s \gt n/2$. The inequality fails for
$s = n/2$. A Sobolev embedding result of Milman and Pustylnik,
originally phrased in a slightly different context, implies an
endpoint inequality: if $n \geq 3$ and $\Omega \subset \mathbb{R}^n$
is bounded, then
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c_n
\left\ \Delta u
\right\_{L^{\frac{n}{2},1}(\Omega)},$$
where $L^{p,q}$ is the Lorentz space refinement of $L^p$. This
inequality fails for $n=2$ and we prove a sharp substitute result:
there exists $c\gt 0$ such that for all $\Omega \subset \mathbb{R}^2$
with finite measure
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c \max_{x \in \Omega} \int_{y \in \Omega}{ \max
\left\{ 1, \log{
\left(\frac{\Omega}{\xy\^2}
\right)}
\right\}
\left \Delta u(y)
\right dy}.$$
This is somewhat dual to the classical TrudingerMoser inequality;
we also note that it is sharper than the usual estimates given
in Orlicz spaces, the proof is rearrangementfree.
The Laplacian can be replaced by any uniformly elliptic operator
in divergence form.
Keywords:AlexandrovBakelmanPucci estimate, second order Sobolev inequality, TrudingerMoser inequality Categories:35A23, 35B50, 28A75, 49Q20 

2. CMB 2016 (vol 60 pp. 104)
3. CMB 2016 (vol 60 pp. 641)
 Werner, Elisabeth; Ye, Deping

Mixed $f$divergence for Multiple Pairs of Measures
In this paper, the concept of the classical $f$divergence for
a pair of measures is extended to the mixed $f$divergence for
multiple pairs of measures. The mixed $f$divergence provides
a way to measure the difference between multiple pairs of (probability)
measures. Properties for the mixed $f$divergence are established,
such as permutation invariance and symmetry in distributions.
An
AlexandrovFenchel type inequality and an isoperimetric inequality
for the
mixed $f$divergence are proved.
Keywords:AlexandrovFenchel inequality, $f$dissimilarity, $f$divergence, isoperimetric inequality Categories:28XX, 52XX, 60XX 

4. CMB 2016 (vol 59 pp. 878)
 Wang, Jianfei

The Carleson Measure Problem Between Analytic Morrey Spaces
The purpose of this paper is to characterize positive measure
$\mu$ on the unit disk such that the analytic
Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly
embedded to the tent space
$\mathcal{T}_{q,1\frac{q}{p}(1\eta)}^{\infty}(\mu)$ for the
case $1\leq q\leq p\lt \infty$
respectively. As an application, these results are used to
establish the boundedness and compactness of integral operators
and multipliers between analytic Morrey spaces.
Keywords:Morrey space, Carleson measure problem, boundedness, compactness Categories:30H35, 28A12, 47B38, 46E15 

5. CMB 2014 (vol 58 pp. 71)
 Ghenciu, Ioana

Limited Sets and Bibasic Sequences
Bibasic sequences are used to study relative weak compactness
and relative norm compactness of limited sets.
Keywords:limited sets, $L$sets, bibasic sequences, the DunfordPettis property Categories:46B20, 46B28, 28B05 

6. CMB 2012 (vol 57 pp. 240)
 Bernardes, Nilson C.

Addendum to ``Limit Sets of Typical Homeomorphisms''
Given an integer $n \geq 3$,
a metrizable compact topological $n$manifold $X$ with boundary,
and a finite positive Borel measure $\mu$ on $X$,
we prove that for the typical homeomorphism $f : X \to X$,
it is true that for $\mu$almost every point $x$ in $X$ the restriction of
$f$ (respectively of $f^{1}$) to the omega limit set $\omega(f,x)$
(respectively to the alpha limit set $\alpha(f,x)$) is topologically
conjugate to the universal odometer.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 

7. CMB 2011 (vol 56 pp. 292)
 Dai, MeiFeng

Quasisymmetrically Minimal Moran Sets
M. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor
sets of Hausdorff dimension $1$, where at the $k$th set one removes
from each interval $I$ a certain number $n_{k}$ of open subintervals
of length $c_{k}I$, leaving $(n_{k}+1)$ closed subintervals of
equal length. Quasisymmetrically Moran sets of Hausdorff dimension $1$
considered in the paper are more general than uniform Cantor sets in
that neither the open subintervals nor the closed subintervals are
required to be of equal length.
Keywords:quasisymmetric, Moran set, Hausdorff dimension Categories:28A80, 54C30 

8. CMB 2011 (vol 56 pp. 326)
9. CMB 2011 (vol 56 pp. 354)
 Hare, Kathryn E.; Mendivil, Franklin; Zuberman, Leandro

The Sizes of Rearrangements of Cantor Sets
A linear Cantor set $C$ with zero Lebesgue measure is associated with
the countable collection of the bounded complementary open intervals. A
rearrangment of $C$ has the same lengths of its complementary
intervals, but with different locations. We study the Hausdorff and packing
$h$measures and dimensional properties of the set of all rearrangments of
some given $C$ for general dimension functions $h$. For each set of
complementary lengths, we construct a Cantor set rearrangement which has the
maximal Hausdorff and the minimal packing $h$premeasure, up to a constant.
We also show that if the packing measure of this Cantor set is positive,
then there is a rearrangement which has infinite packing measure.
Keywords:Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cutout set Categories:28A78, 28A80 

10. CMB 2011 (vol 55 pp. 830)
 Reinhold, Karin; Savvopoulou, Anna K.; Wedrychowicz, Christopher M.

Almost Everywhere Convergence of Convolution Measures
Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $(X,\mathcal{B},m)$ a probability
space and $\tau$ an invertible, measure preserving transformation.
This paper deals with the almost everywhere convergence in $\textrm{L}^1(X)$ of a
sequence of operators of weighted averages. Almost everywhere convergence follows
once we obtain an appropriate maximal estimate and once we provide
a dense class where convergence holds almost everywhere.
The weights are given by convolution products of members of a sequence of probability
measures $\{\nu_i\}$ defined on $\mathbb{Z}$.
We then exhibit cases of such averages where convergence fails.
Category:28D 

11. CMB 2011 (vol 55 pp. 723)
 Gigli, Nicola; Ohta, ShinIchi

First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces
We extend results proved by the second author (Amer. J. Math., 2009)
for nonnegatively curved Alexandrov spaces
to general compact Alexandrov spaces $X$ with curvature bounded
below.
The gradient flow of a geodesically convex functional on the quadratic Wasserstein
space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality.
Moreover, the gradient flow enjoys uniqueness and contractivity.
These results are obtained by proving a first variation formula for
the Wasserstein distance.
Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow Categories:53C23, 28A35, 49Q20, 58A35 

12. CMB 2011 (vol 55 pp. 225)
 Bernardes, Nilson C.

Limit Sets of Typical Homeomorphisms
Given an integer $n \geq 3$, a metrizable compact
topological $n$manifold $X$ with boundary, and a finite positive Borel
measure $\mu$ on $X$, we prove that for the typical homeomorphism
$f \colon X \to X$, it is true that for $\mu$almost every point $x$ in $X$
the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero,
each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is
nonsensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 

13. CMB 2011 (vol 55 pp. 815)
 Oberlin, Daniel M.

Restricted Radon Transforms and Projections of Planar Sets
We establish a mixed norm estimate for the Radon transform in
$\mathbb{R}^2$ when the set of directions has fractional dimension.
This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to
a conjecture analogous to a wellknown conjecture of Furstenberg.
Categories:44A12, 28A78 

14. CMB 2011 (vol 54 pp. 706)
15. CMB 2010 (vol 54 pp. 172)
 Shayya, Bassam

Measures with Fourier Transforms in $L^2$ of a Halfspace
We prove that if the Fourier transform of a compactly supported
measure is in $L^2$ of a halfspace, then the measure is
absolutely continuous to Lebesgue measure. We then show how this
result can be used to translate information about the
dimensionality of a measure and the decay of its Fourier
transform into geometric information about its support.
Categories:42B10, 28A75 

16. CMB 2009 (vol 40 pp. 3)
 Ayari, S.; Dubuc, S.

La formule de Cauchy sur la longueur d'une courbe
Pour toute courbe rectifiable du plan, nous d\'emontrons la formule
de Cauchy relative \`a sa longueur. La formule est donn\'ee sous deux
formes: comme int\'egrale de la variation totale des projections de la
courbe dans les diverses directions et comme int\'egrale double du
nombre de rencontres de la courbe avec une droite quelconque du plan.
We give a general proof of the Cauchy formula about the length of a
plane curve. The formula is given in two ways: as the integral of the
variation of orthogonal projections of the curve, and as a double
integral of the number of intersections of the curve with an arbitrary
line of the plane.
Keywords:longueur, variation bornÃ©e, gÃ©omÃ©trie intÃ©grale Categories:28A75, 28A45 

17. CMB 2009 (vol 52 pp. 105)
18. CMB 2006 (vol 49 pp. 247)
19. CMB 2006 (vol 49 pp. 203)
 Çömez, Doğan

The Ergodic Hilbert Transform for Admissible Processes
It is shown that the ergodic Hilbert transform
exists for a class of bounded symmetric admissible processes
relative to invertible measure preserving transformations. This
generalizes the wellknown result on the existence of the ergodic
Hilbert transform.
Keywords:Hilbert transform, admissible processes Categories:28D05, 37A99 

20. CMB 2004 (vol 47 pp. 168)
 Baake, Michael; Sing, Bernd

Kolakoski$(3,1)$ Is a (Deformed) Model Set
Unlike the (classical) Kolakoski sequence on the alphabet $\{1,2\}$, its analogue
on $\{1,3\}$ can be related to a primitive substitution rule. Using this connection,
we prove that the corresponding biinfinite fixed point is a regular generic model set
and thus has a pure point diffraction spectrum. The Kolakoski$(3,1)$ sequence is
then obtained as a deformation, without losing the pure point diffraction property.
Categories:52C23, 37B10, 28A80, 43A25 

21. CMB 2002 (vol 45 pp. 123)
 Moody, Robert V.

Uniform Distribution in Model Sets
We give a new measuretheoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 

22. CMB 2001 (vol 44 pp. 429)
 Henniger, J. P.

Ergodic Rotations of Nilmanifolds Conjugate to Their Inverses
In answer to a question posed in \cite{G}, we give sufficient
conditions on a Lie nilmanifold so that any ergodic rotation of the
nilmanifold is metrically conjugate to its inverse. The condition is
that the Lie algebra be what we call quasigraded, and is weaker than
the property of being graded. Furthermore, the conjugating map can be
chosen to be an involution. It is shown that for a special class of
groups, the condition of quasigraded is also necessary. In certain
examples there is a continuum of conjugacies.
Categories:28Dxx, 22E25 

23. CMB 2001 (vol 44 pp. 61)
 Kats, B. A.

The Inequalities for Polynomials and Integration over Fractal Arcs
The paper is dealing with determination of the integral $\int_{\gamma}
f \,dz$ along the fractal arc $\gamma$ on the complex plane by terms
of polynomial approximations of the function~$f$. We obtain
inequalities for polynomials and conditions of integrability for
functions from the H\"older, Besov and Slobodetskii spaces.
Categories:26B15, 28A80 

24. CMB 2000 (vol 43 pp. 157)
25. CMB 1999 (vol 42 pp. 291)
 Grubb, D. J.; LaBerge, Tim

Spaces of QuasiMeasures
We give a direct proof that the space of Baire quasimeasures on a
completely regular space (or the space of Borel quasimeasures on a
normal space) is compact Hausdorff. We show that it is possible for
the space of Borel quasimeasures on a nonnormal space to be
noncompact. This result also provides an example of a Baire
quasimeasure that has no extension to a Borel quasimeasure. Finally,
we give a concise proof of the WheelerShakmatov theorem, which states
that if $X$ is normal and $\dim(X) \le 1$, then every
quasimeasure on $X$ extends to a measure.
Category:28C15 
