On the gentle properties of anisotropic Besov spaces

In this paper, we prove that anisotropic homogeneous Besov spaces ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ are gentle spaces, for all parameters $s,p,q$ and all anisotropies $u$. Using the Littlewood–Paley decomposition, we study their completeness, separability, duality and homogeneity. We then define the notion of anisotropic orthonormal wavelet basis of $L^2\left({\mathbb{R}}^d\right)$, and we show that the homogeneous version of Triebel families of anisotropic orthonormal wavelet bases associated to the tensor product of Lemarié–Meyer (resp. Daubechies) wavelets are particular examples. We characterize the ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ spaces using Lemarié–Meyer wavelets. In fact, we show that these bases will be either unconditional bases or unconditional $1$-weak bases of ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$, depending on whether ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ is separable or not. By introducing an anisotropic version of the class of almost diagonal matrices related to anisotropic orthonormal wavelet bases, we prove that these spaces are stable under changes of anisotropic orthonormal wavelet bases. As a consequence, we extend the characterization of ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ using Daubechies wavelets.     

On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

Two remarks on remotality

We prove that there exists a weakly closed and bounded subset $E$ of $c_0$ which is not remotal from 0, and such that $\overline{\mathrm{co}}\left(E\right)$ is remotal from 0. This answers a question of M. Martín and T.S.S.R.K. Rao. We also present a simple proof of the fact that in every non-reflexive Banach space there exists a closed convex bounded set which is not remotal.     

Bipartite algebraic graphs without quadrilaterals

Let ${\mathbb{P}}^s$ be the $s$-dimensional complex projective space, and let $X,Y$ be two non-empty open subsets of ${\mathbb{P}}^s$ in the Zariski topology. A hypersurface $H$ in ${\mathbb{P}}^s\times {\mathbb{P}}^s$ induces a bipartite graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set is defined by $\overline{u}～\overline{v}$ if and only if $(\overline{u},\overline{v})\in H$. Motivated by the Turán problem for bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$ and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\overline{y}$ is bounded by a constant $d=d(s,t)$, and we discuss possible notions of the equivalence.We establish the result that if $H\cap (X\times {\mathbb{P}}^2)$ is $(2,2)$-grid-free, then there exists $F\in \mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such that $H\cap (X\times {\mathbb{P}}^2)=\{F=0\}\cap (X\times {\mathbb{P}}^2)$. Finally, we transfer the result to algebraically closed fields of large characteristic.