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.     

Three new results on continuation criteria for the 3D relativistic Vlasov–Maxwell system

We consider continuation criteria for the three-dimensional relativistic Vlasov–Maxwell system. When the particle density, $f(t,x,p)$, is compactly supported at $t=0$, we prove ${6p_0^{\frac{18}{5r}?1+\beta }f6}_{L_t^{\infty }{L}_x^r{L}_p^1}?1$, where $1\le r\le 2$ and $\beta >0$ is arbitrarily small, is a continuation criteria. Our continuation criteria is an improvement in the $1\le r\le 2$ range to the previously best known criteria ${6p_0^{\frac{4}{r}?1+\beta }f6}_{L_t^{\infty }{L}_x^r{L^1}_p}?1$ due to Kunze [7]. We also consider continuation criteria when $f(0,x,p)$ has noncompact support. In this regime, Luk–Strain [9] proved that ${6p_0^{\theta }f6}_{L_x^1{L}_p^1}?1$ is a continuation criteria for $\theta >5$. We improve this result to $\theta >3$. Finally, we build on another result by Luk–Strain [8]. The authors proved boundedness of momentum support on a fixed two-dimensional plane is a sufficient continuation criteria. We prove the same result even if the plane varies continuously in time.     

The zero-one law for a complete orthonormal system

A complete orthonormal system of functions $\Theta ={\left\{{\theta }_n\right\}}_{n=1}^{\infty }\text{,}{\theta }_n\in L_{[0\text{,}1]}^{\infty }$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0\text{,}1]$ if ${\left\{a_n\right\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a.e. for any ${\left\{a_n\right\}}_{n=1}^{\infty }?l^2$. We also show that for any complete ONS ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions defined on $[0\text{,}1]$ there exists a fixed non decreasing subsequence ${\left\{n_k\right\}}_{k=1}^{\infty }$ of natural numbers such that for any $f\in L_{[0\text{,}1]}^0$ and some sequence of coefficients ${\left\{b_n\right\}}_{n=1}^{\infty }$,

$\sum _{n=1}^{n_k}{b}_n{f}_n\to f\text{a.e. when}k\to \infty \text{.}$

To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On the sum of reciprocals of least common multiples

Let ${\{a_i\}}_{i=1}^{\infty }$ be a strictly increasing sequence of positive integers ($a_i<a_j$ if $i<j$). In 1978, Borwein showed that for any positive integer n, we have ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,a_{i+1})}\le 1?\frac{1}{2^n}$, with equality occurring if and only if $a_i=2^{i?1}$ for $1\le i\le n+1$. Let $3\le r\le 7$ be an integer. In this paper, we investigate the sum ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}$ and show that ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}\le U_r(n)$ for any positive integer n, where $U_r(n)$ is a constant depending on r and n. Further, for any integer $n\ge 2$, we also give a characterization of the sequence ${\{a_i\}}_{i=1}^{\infty }$ such that the equality ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}=U_r(n)$ holds.     

Lp- and Sp,qrB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base $b$. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness $S_{p,q}^{r}B\left({[0,1)}^s\right)$, which will also give us bounds on the $L_p$-discrepancy. Our sequence and point sets will achieve the known optimal order for the $L_p$- and $S_{p,q}^{r}B$-discrepancy. The results in this paper generalize several previous results on $L_p$- and $S_{p,q}^{r}B$-discrepancy estimates and provide a sharp upper bound on the $S_{p,q}^{r}B$-discrepancy of one-dimensional sequences for $r>0$. We will use the $b$-adic Haar function system in the proofs.     

Mordell type exponential sum estimates in fields of prime order

We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials $f(x)={\sum }_{i=1}^r{a}_i{x}^{k_i}\text{,}(a_i\text{,}p)=1\text{,}p$ prime, under essentially optimal conditions on the exponents $1?k_i<p?1$. The method is based on sum–product estimates in finite fields ${\mathbb{F}}_p$ and their Cartesian products. We also obtain estimates on incomplete sums of the form ${\sum }_{s=1}^t{e}_p({\sum }_{i=1}^r{a}_i{\theta }_i^s)$ for $t>p^{?}$, under appropriate conditions on the ${\theta }_i\in {\mathbb{F}}_p^{*}$. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On two Diophantine inequalities over primes

Let $1<c<37∕18,c\ne 2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in (N,2N]$, the Diophantine inequality $|p_1^c+p_2^c+p_3^c?R|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3$. Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c?N|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3,p_4,p_5,p_6$ for sufficiently large real number $N$.     

Translational absolute continuity and Fourier frames on a sum of singular measures

A finite Borel measure μ in ${\mathbb{R}}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(\mu )$. It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures ν and λ without atoms whose supports form a packing pair, then $\nu ?\lambda +{\delta }_t?\nu$ is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure ${\mu }_4+{\mu }_{16}$ does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping $R_{\theta }$ with $\theta \ne \pm \pi /2$, the two-dimensional measure ${\rho }_{\theta }(?):=({\mu }_4\times {\delta }_0)(?)+({\delta }_0\times {\mu }_{16})(R_{\theta }^{?1}?)$, supported on the union of x-axis and $y=(\cot ?\theta )x$, always admit a Fourier frame. Furthermore, we can find ${\{e^{2\pi i〈\lambda ,x〉}\}}_{\lambda \in {\mathrm{\Lambda }}_{\theta }}$ such that it forms a Fourier frame for ${\rho }_{\theta }$ with frame bounds independent of θ. Nonetheless, ${\rho }_{\pm \pi /2}$ does not admit any Fourier frame.