In this paper, we look at various arithmetic properties of the set of those positive integers $n$ whose sum of digits in a fixed base $b>1$ is a fixed positive integers $s$. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler $\phi$ function.

In this paper we study the $5$ families of genus $3$ compact Riemann surfaces which are normal coverings of the Riemann sphere branched over $4$ points from very different aspects: their moduli spaces, the uniform Belyi functions that factorize through the quotient by the automorphism groups and the Weierstrass points of the non hyperelliptic families.

We characterize the set of all functions $f$ of $\mathbb R$ to itself such that the associated superposition operator $T_f: g \to f \circ g$ maps the class $BV^1_p (\mathbb R)$ into itself. Here $BV^1_p (\mathbb R)$, $1 \le p < \infty$, denotes the set of primitives of functions of bounded $p$-variation, endowed with a suitable norm. It turns out that such an operator is always bounded and sublinear. Also, consequences for the boundedness of superposition operators defined on Besov spaces $B^s_{p,q}({\mathbb R}^n)$ are discussed.

In this paper, we study the singular vortex patches in the two-dimensional incompressible Navier-Stokes equations. We show, in particular, that if the initial vortex patch is $C^{1+s}$ outside a singular set $\Sigma$, so the velocity is, for all time, lipschitzian outside the image of $\Sigma$ through the viscous flow. In addition, the correponding lipschitzian norm is independant of the viscosity. This allows us to prove some results related to the inviscid limit for the geometric structures of the vortex patch.

In this note we provide a natural way of defining exponential coordinates on the class of probabilities on the set $\Omega = [1,n]$ or on $\mathbb{P} = \{p=(p_1,\dots,p_n)\in \mathbb{R}^n | p_i > 0; \Sigma_{i=1}^n p_i =1\}$. For that we have to regard $\mathbb{P}$ as a projective space and the exponential coordinates will be related to geodesic flows in $\mathbb{C}^n$.

In this paper, we are concerned with the asymptotically linear elliptic problem $-\Delta u+ \lambda_{0}u=f(u), u\in H_{0}^{1}(\Omega ) $ in an exterior domain $\Omega= \mathbb{R}^{N}\setminus\overline{\mathcal{O}} \left( N\geqslant 3\right) $ with $\mathcal{O}$ a smooth bounded and star-shaped open set, and $\lim_{t\rightarrow +\infty }\frac{ f(t)}{t}=l$, $0<l<+\infty$. Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution.

Let $(M, g)$ be a complete Riemannian manifold, $L=\Delta -\nabla \phi \cdot \nabla$ be a Markovian symmetric diffusion operator with an invariant measure $d\mu(x)=e^{-\phi(x)}d\nu(x)$, where $\phi\in C^2(M)$, $\nu$ is the Riemannian volume measure on $(M, g)$. A fundamental question in harmonic analysis and potential theory asks whether or not the Riesz transform $R_a(L)=\nabla(a-L)^{-1/2}$ is bounded in $L^p(\mu)$ for all $1<p<\infty$ and for certain $a\geq 0$. An affirmative answer to this problem has many important applications in elliptic or parabolic PDEs, potential theory, probability theory, the $L^p$-Hodge decomposition theory and in the study of Navier-Stokes equations and boundary value problems. Using some new interplays between harmonic analysis, differential geometry and probability theory, we prove that the Riesz transform $R_a(L)=\nabla(a-L)^{-1/2}$ is bounded in $L^p(\mu)$ for all $a>0$ and $p\geq 2$ provided that $L$ generates a ultracontractive Markovian semigroup $P_t=e^{tL}$ in the sense that $P_t 1=1$ for all $t\geq 0$, $\|P_t\|_{1, \infty} < Ct^{-n/2}$ for all $t\in (0, 1]$ for some constants $C>0$ and $n > 1$, and satisfies $$ (K+c)^{-}\in L^{{n\over 2}+\epsilon}(M, \mu) $$ for some constants $c\geq 0$ and $\epsilon>0$, where $K(x)$ denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature $Ric(L)=Ric+\nabla^2\phi$ on $T_x M$, i.e., $$ K(x)=\inf\limits\{Ric(L)(v, v): v\in T_x M, \|v\|=1\}, \quad\forall\ x\in M. $$ Examples of diffusion operators on complete non-compact Riemannian manifolds with unbounded negative Ricci curvature or Bakry-Emery Ricci curvature are given for which the Riesz transform $R_a(L)$ is bounded in $L^p(\mu)$ for all $p\geq 2$ and for all $a>0$ (or even for all $a\geq 0$).

In this paper, using a recent parabolic restriction estimate of Tao, we obtain improved partial results in the direction of Falconer's distance set conjecture in dimensions $d\geq 3$.

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.

We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls.

We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand's class, with methods of time-frequency analysis (phase space analysis). Compared to the classical treatment, the time-frequency approach leads to striklingly simple proofs of Sjöstrand's fundamental results and to far-reaching generalizations.

On établit l'estimation sous-linéaire $\|f \circ g\| \leq c(f) \|g\|$, la norme étant celle de l'espace de Besov homogène $\dot B^{s,q}_{p}(\mathbb{R}^n)$, où $1\leq p<+\infty$, $1\leq q\leq +\infty$ et $0<s<1+(1/p)$. La fonction $f$ est supposée appartenir à une classe $U^1_p$ introduite antérieurement par Bourdaud et Kateb, qui contient notamment les primitives des fonctions à variation bornée sur ${\mathbb R}$. À titre d'application, on montre qu'une fonction étagée appartient à $\dot{B}^{1,\infty}_1({\mathbb R^2})$ si et seulement si elle appartient à $ BV({\mathbb R^2})$.