We study the focusing 3d cubic NLS equation with $H^1$ data at the mass-energy threshold, namely, when $M[u_0]E[u_0]{=}M[Q]E[Q]$. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when $M[u_0]E[u_0] < M[Q]E[Q]$. In this paper, we first exhibit 3 special solutions: $e^{it} Q$ and $Q^\pm$, where $Q$ is the ground state, $Q^\pm$ exponentially approach the ground state solution in the positive time direction, $Q^+$ has finite time blow up and $Q^-$ scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to $\dot{H}^{1/2}$ symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.

We consider holomorphic self-maps $\varphi$ of the unit ball $\mathbb B^N$ in $\mathbb C^N$ ($N=1,2,3,\dots$). In the one-dimensional case, when $\varphi$ has no fixed points in $\mathbb D\defeq \mathbb B^1$ and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map $\varphi$, and therefore, in this case, the dynamical properties of $\varphi$ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on $\varphi$ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation $\sigma$, which maps the ball into the right half-plane of $\mathbb C$, and solves the functional equation $\sigma\circ \varphi=\lambda \sigma$, where $\lambda > 1$ is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of $\varphi$.

The functional class of Hölder exponents of continuous function has been completely characterized by P. Andersson, K. Daoudi, S. Jaffard, J. Lévy-Véhel and Y. Meyer [Andersson, P.: Wavelets and local regularity. PhD Thesis. Department of Mathematics, G\"oteborg, 1997], [Andersson, P.: Characterization of pointwise Hölder regularity. Appl. Comput. Harmon. Anal. {\bf 4} (1997), 429-443], [Daoudi, K., Lévy-Véhel J. and Meyer, Y.: Construction of continuous functions with prescribed local regularity. Constr. Approx. {\bf 14} (1998), 349-385], [Jaffard, S.: Functions with prescribed Hölder exponent. Appl. Comput. Harmon. Anal. {\bf 2} (1995), 400-401]; these authors have shown that this class exactly corresponds to that of the lower limits of the sequences of nonnegative continuous functions. The problem of determining whether or not the Hölder exponents of discontinuous (and even unbounded) functions can belong to a larger class remained open during the last decade. The main goal of our article is to show that this is not the case: the latter Hölder exponents can also be expressed as lower limits of sequences of continuous functions. Our proof mainly relies on a ``wavelet-leader'' reformulation of a nice characterization of pointwise Hölder regularity due to P. Anderson.

We prove almost sharp mixed-norm estimates for the $X$-ray transform restricted to lines whose directions lie on certain well-curved two dimensional manifolds.

In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian.

Suppose that $f$ is a function on $\mathbb{R}^n$ such that $\exp(a |\cdot|^2) f$ and $\exp(b |\cdot|^2) \hat f$ are bounded, where $a,b > 0$. Hardy's Uncertainty Principle asserts that if $ab > \pi^2$, then $f = 0$, while if $ab = \pi^2$, then $f = c\exp(-a|\cdot|^2)$. In this paper, we generalise this uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.

We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have $L^q$-Lebesgue points everywhere.

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f\sb\lambda)\subseteq \mathscr{D}'(\Omega)$ of distributions depending in a real analytic way on $\lambda\in U$, $U$ a real analytic manifold, there is a family of solutions $(u\sb\lambda)\subseteq \dio$ also depending analytically on $\lambda$ such that $$ P(D)u\sb\lambda=f\sb\lambda\qquad \text{for every $\lambda\in U$}, $$ where $\Omega\subseteq \mathbb{R}\sp d$ an open set. For general surjective variable coefficients operators or operators acting on currents over a smooth manifold we give a solution in terms of an abstract ``Hadamard three circle property'' for the kernel of the operator. The obtained condition is evaluated giving the full solution (usually in terms of the symbol) for operators with constant coefficients and open (convex) $\Omega\subseteq\mathbb{R}\sp d$ if $P(D)$ is of one of the following types: 1) elliptic, 2) hypoelliptic, 3) homogeneous, 4) of two variables, 5) of order two or 6) if $P(D)$ is the system of Cauchy-Riemann equations. An analogous problem is solved for convolution operators of one variable. In all enumerated cases, it follows that the solution is in the affirmative if and only if $P(D)$ has a linear continuous right inverse which shows a striking difference comparing with analogous smooth or holomorphic parameter dependence problems. The paper contains the whole theory working also for operators on Beurling ultradistributions $\mathscr{D}'\sb{(\omega)}$. We prove a characterization of surjectivity of tensor products of general surjective linear operators on a wide class of spaces containing most of the natural spaces of classical analysis.

In the setting of step two Carnot groups we give an explicit representation of Taylor polynomial in terms of a suitable basis of the real vector space of left invariant differential operators, acting pointwisely on monomials like the ordinary Euclidean iterated derivations.

We consider an aggregation equation in $\mathbb R^d$, $d\ge 2$ with fractional dissipation: $u_t + \nabla\cdot(u \nabla K*u)=-\nu \Lambda^\gamma u $, where $\nu\ge 0$, $0 < \gamma\le 2$ and $K(x)=e^{-|x|}$. In the supercritical case, $0 < \gamma < 1$, we obtain new local wellposedness results and smoothing properties of solutions. In the critical case, $\gamma=1$, we prove the global wellposedness for initial data having a small $L_x^1$ norm. In the subcritical case, $\gamma > 1$, we prove global wellposedness and smoothing of solutions with general $L_x^1$ initial data.

We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the $L_x^\infty$-norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczy\'nski [Biler, P. and Woyczy\'Nski, W.A.: Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math. {\bf 59} (1999), no. 3, 845-869].

We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique.

We present a result of $L^p$ continuity of singular integrals of Calderón-Zygmund type in the context of bounded nonhomogeneous spaces, well suited to be applied to problems of a priori estimates for partial differential equations. First, an easy and selfcontained proof of $L^2$ continuity is got by means of $C^{\alpha}$ continuity, thanks to an abstract theorem of Krein. Then $L^p$ continuity is derived adapting known results by Nazarov-Treil-Volberg about singular integrals in nonhomogeneous spaces.