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1. CMB Online first

Steinerberger, Stefan
An Endpoint Alexandrov Bakelman Pucci estimate in the Plane
The classical Alexandrov-Bakelman-Pucci 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 $$ \max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|} + c \max_{x \in \Omega} \int_{y \in \Omega}{ \max \left\{ 1, \log{ \left(\frac{|\Omega|}{\|x-y\|^2} \right)} \right\} \left| \Delta u(y) \right| dy}.$$ This is somewhat dual to the classical Trudinger-Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces, the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.

Keywords:Alexandrov-Bakelman-Pucci estimate, second order Sobolev inequality, Trudinger-Moser inequality
Categories:35A23, 35B50, 28A75, 49Q20

2. CMB 2016 (vol 60 pp. 104)

Diestel, Geoff
An Extension of Nikishin's Factorization Theorem
A Nikishin-Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow.

Keywords:factorization, type, cotype, Banach spaces
Categories:46E30, 28A25

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 Alexandrov-Fenchel type inequality and an isoperimetric inequality for the mixed $f$-divergence are proved.

Keywords:Alexandrov-Fenchel inequality, $f$-dissimilarity, $f$-divergence, isoperimetric inequality
Categories:28-XX, 52-XX, 60-XX

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 Dunford-Pettis 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, Mei-Feng
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)

Erdoğan, M. Burak; Oberlin, Daniel M.
Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$
We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms of some fractal measures.

Keywords:Fourier transforms of fractal measures, Fourier restriction
Categories:42B10, 28A12

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, cut-out 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.


11. CMB 2011 (vol 55 pp. 723)

Gigli, Nicola; Ohta, Shin-Ichi
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 non-sensitive 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 well-known conjecture of Furstenberg.

Categories:44A12, 28A78

14. CMB 2011 (vol 54 pp. 706)

Moonens, Laurent
Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve
We provide a simple example showing that the tangential derivative of a continuous function $\phi$ can vanish everywhere along a curve while the variation of $\phi$ along this curve is nonzero. We give additional regularity conditions on the curve and/or the function that prevent this from happening.

Categories:26A24, 28A15

15. CMB 2010 (vol 54 pp. 172)

Shayya, Bassam
Measures with Fourier Transforms in $L^2$ of a Half-space
We prove that if the Fourier transform of a compactly supported measure is in $L^2$ of a half-space, 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 bornée, géométrie intégrale
Categories:28A75, 28A45

17. CMB 2009 (vol 52 pp. 105)

Okoudjou, Kasso A.; Rogers, Luke G.; Strichartz, Robert S.
Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket
We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet.

Categories:28A80, 31C45

18. CMB 2006 (vol 49 pp. 247)

Myjak, Józef; Szarek, Tomasz; Ślȩczka, Maciej
A Szpilrajn--Marczewski Type Theorem for Concentration Dimension on Polish Spaces
Let $X$ be a Polish space. We will prove that $$ \dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to } X\}, $$ where $\dim_L X$ and $\dim_T X$ stand for the concentration dimension and the topological dimension of $X$, respectively.

Keywords:Hausdorff dimension, topological dimension, Lévy concentration function, concentration dimension
Categories:11K55, 28A78

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 well-known 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 bi-infinite 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 measure-theoretical 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 quasi-graded, 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 quasi-graded 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)

El Abdalaoui, El Houcein
A Larger Class of Ornstein Transformations with Mixing Property
We prove that Ornstein transformations are almost surely totally ergodic provided only that the cutting parameter is not bounded. We thus obtain a larger class of Ornstein transformations with the mixing property.

Categories:28D05, 47A35

25. CMB 1999 (vol 42 pp. 291)

Grubb, D. J.; LaBerge, Tim
Spaces of Quasi-Measures
We give a direct proof that the space of Baire quasi-measures on a completely regular space (or the space of Borel quasi-measures on a normal space) is compact Hausdorff. We show that it is possible for the space of Borel quasi-measures on a non-normal space to be non-compact. This result also provides an example of a Baire quasi-measure that has no extension to a Borel quasi-measure. Finally, we give a concise proof of the Wheeler-Shakmatov theorem, which states that if $X$ is normal and $\dim(X) \le 1$, then every quasi-measure on $X$ extends to a measure.

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