Un contre-exemple pour un espace d'interpolation qui n'est pas faiblement LUR

According to a previous result of the author, if $(A_0,A_1)$ is an interpolation couple, if $A_0^{?}$ is weakly LUR, then the complex interpolation spaces ${(A_0^{?},A_1^{?})}_{\theta }$ have the same property.Here we construct an interpolation couple $(B_0,B_1)$ where $B_0$ is LUR, but where the complex interpolation spaces ${(B_0,B_1)}_{\theta }$ are not strictly convex.     

Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field

${\epsilon }^{2s}{(?\mathrm{\Delta })}_{A/\epsilon }^{s}u+V(x)u=f(|u{|}^2)u\text{ in }{\mathbb{R}}^N,$

where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N\ge 3$, ${(?\mathrm{\Delta })}_A^s$ is the fractional magnetic Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ are continuous potentials and $f:{\mathbb{R}}^N\to \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for ε small.     

Dedekind–Mertens lemma and content formulas in power series rings

Let $R?X?$ be the power series ring over a commutative ring R with identity. For $f\in R?X?$, let $A_f$ denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if $g\in R?X?$ such that $A_g$ is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely $A_f^2{A}_g=A_f{A}_{fg}$ for all $f\in R?X?$. More generally for a Prüfer domain R, we prove the content formula ${(A_f{A}_g)}^2=(A_f{A}_g)A_{fg}$ for all $f,g\in R?X?$. As a consequence it is shown that an integral domain R is completely integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R?X?$, which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R[X]$, where $R[X]$ is the polynomial ring over R.For a ring R and $g\in R?X?$, if $A_g$ is not locally finitely generated, then there may be no positive integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. Assuming that the locally minimal number of generators of $A_g$ is $k+1$, Epstein and Shapiro posed a question about the validation of the formula $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of $A_g$ is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of $A_g$ is $k+1$, then $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. As a consequence we show that if $A_g$ is finitely generated (in particular if $g\in R[X]$), then there exists a nonnegative integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$.     

Ideals in deformation quantizations over Z/pnZ

Let k be a perfect field of characteristic $p>2$. Let $A_1$ be an Azumaya algebra over a smooth symplectic affine variety over k. Let $A_n$ be a deformation quantization of $A_1$ over $W_n(\mathbf{k})$. We prove that all $W_n(\mathbf{k})$-flat two-sided ideals of $A_n$ are generated by central elements.     

Superconcentration,and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in ${\mathbb{R}}^n$in the ?-position, and such that the space $({\mathbb{R}}^n,{6?6}_B)$ admits a 1-unconditional basis. Then for any $\epsilon \in (0,1/2]$, and for random $c\epsilon \log ?n/\log ?\frac{1}{\epsilon }$-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\epsilon )$-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ?-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and ${\mathbb{E}}_{{\gamma }_n}{6?6}_B\le n^c{\mathbb{E}}_{{\gamma }_n}{6{\mathrm{grad}}_B(?)6}_2$ for a small universal constant $c>0$, where ${\mathrm{grad}}_B(?)$ is the gradient of ${6?6}_B$ and ${\gamma }_n$ is the standard Gaussian measure in ${\mathbb{R}}^n$. Then for any $p\in [1,c\log ?n]$ the p-th power of the norm ${6?6}_B^p$ is $\frac{C}{\log ?n}$-superconcentrated in the Gauss space.