In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method is simpler and is of independent interest since it mainly relies on general properties of finite sets of points satisfying a scale-invariant flatness condition. For instance it shows that a quasi-arc of small constant cannot contain "sharp saw-teeth".

The infinite Brownian loop $\{B_t^0,t\ge 0\}$ on a Riemannian manifold $\mathbb{M}$ is the limit in distribution of the Brownian bridge of length $T$ around a fixed origin $0$, when $T\to+\infty$. It has no spectral gap. When $\mathbb{M}$ has nonnegative Ricci curvature, $B^0$ is the Brownian motion itself. When $\mathbb{M}=G/K$ is a noncompact symmetric space, $B^0$ is the relativized $\Phi_0$-process of the Brownian motion, where $\Phi_0$ denotes the basic spherical function of Harish-Chandra, i.e. the $K$-invariant ground state of the Laplacian. In this case, we consider the polar decomposition $B_t^0=(K_t,X_t)$, where $K_t\in K/M$ and $X_t\in\conec$, the positive Weyl chamber. Then, as $t\to+\infty$, $K_t$ converges and $d(0,X_t)/t\to0$ almost surely. Moreover the processes $\{X_{tT}/\sqrt{T},t\ge 0\}$ converge in distribution, as $T\to+\infty$, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that $d(0,X_{tT})/\sqrt{T}$ converges to a Bessel process of dimension $D=rank \mathbb{M}+2j$, where $j$ denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on $\Phi_0$.

There is a one parameter family of bilinear Hilbert transforms. Recently, some progress has been made to prove $L^p$ estimates for these operators uniformly in the parameter. In the current article we present some of these techniques in a simplified model: we prove the estimate $L^2\times L^2\to L^{1,\infty}$ uniformly for a discrete family of model operators for the bilinear Hilbert transform.

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A typical example of energy functional we consider is the one given by the nonparametric area integrand $f(x, \xi) = \sqrt{1 + \Vert \xi \Vert^2}$, which corresponds with the time-dependent minimal surface equation. We also study the asymptotic behaviour of the solutions.

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the quadratic aspect) in the limit, then, the lack of $L^{2}$ bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.

The concept of uniform convexity of a Banach space was generalized to linear operators between Banach spaces and studied by Beauzamy. Under this generalization, a Banach space $X$ is uniformly convex if and only if its identity map $I_X$ is. Pisier showed that uniformly convex Banach spaces have martingale type $p$ for some $p>1$. We show that this fact is in general not true for linear operators. To remedy the situation, we introduce the new concept of martingale subtype and show, that it is equivalent, also in the operator case, to the existence of an equivalent uniformly convex norm on $X$. In the case of identity maps it is also equivalent to having martingale type $p$ for some $p>1$. Our main method is to use sequences of ideal norms defined on the class of all linear operators and to study the factorization of the finite summation operators. There is a certain analogy with the theory of Rademacher type.