Using Bellman function techniques, we obtain the optimal dependence of the operator norms in $L^2(\mathbb{R})$ of the Haar multipliers $T_w^t$ on the corresponding $RH^d_2$ or $A^d_2$ characteristic of the weight $w$, for $t=1,\pm 1/2$. These results can be viewed as particular cases of estimates on homogeneous spaces $L^2(vd\sigma)$, for $\sigma$ a doubling positive measure and $v\in A^d_2(d\sigma)$, of the weighted dyadic square function $S_{\sigma}^d$. We show that the operator norms of such square functions in $L^2(v d\sigma)$ are bounded by a linear function of the $A^d_2(d\sigma )$ characteristic of the weight $v$, where the constant depends only on the doubling constant of the measure $\sigma$. We also show an inverse estimate for $S_{\sigma}^d$. Both results are known when $d\sigma=dx$. We deduce both estimates from an estimate for the Haar multiplier $(T_v^{\sigma})^{1/2}$ on $L^2(d\sigma)$ when $v\in A_2^d(d\sigma)$, which mirrors the estimate for $T_w^{1/2}$ in $L^2(\mathbb{R})$ when $w\in A^d_2$. The estimate for the Haar multiplier adapted to the $\sigma$ measure, $(T_v^{\sigma})^{1/2}$, is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all $\sigma$, since the particular case $d\sigma=dx$, $v=w$, correspond to the estimates for the Haar multipliers $T^{1/2}_w$ proven to be sharp.

We study group gradings on the Albert algebra and on the exceptional simple Lie algebra $\frak{f}_4$ over algebraically closed fields of characteristic zero. The immediate precedent of this work is [Draper, C. and Martin, C.: Gradings on $\frak{g}_2$. Linear Algebra Appl. 418 (2006), no. 1, 85-111] where we described (up to equivalence) all the gradings on the exceptional simple Lie algebra $\frak{g}_2$. In the cases of the Albert algebra and $\frak{f}_4$, we look for the nontoral gradings finding that there are only eight nontoral nonequivalent gradings on the Albert algebra (three of them being fine) and nine on $\frak{f}_4$ (also three of them fine).

In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of [Martel, Y. and Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157 (2001), 219-254], [Martel, Y. and Merle, F.: Asymptotic stability of solitons of the gKdV equations with a general nonlinearity. Math. Ann. 341 (2008), 391-427]. As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons.

Following old ideas of V. Yu. Krylov we consider the possibility that high order differential operators of dissipative type and constant coefficients might be associated, at least formally, with signed measures on path space in the same way that Wiener measure is associated with the Laplacian. There are fundamental difficulties with this idea because the measure would always have locally infinite mass. However, this paper provides evidence that if one considers equivalence classes of paths corresponding to distinct parameterisations of the same path, the measures might really exist on this quotient space. Precisely, we consider the measures on piecewise linear paths with given time partition defined using the semigroup associated to the differential operator and prove that these measures converge in distribution when the test functions on path space are the iterated integrals of the paths. Given a "random" piecewise-linear path, we evaluate its "expected" signature in terms of an explicit tensor series in the tensor algebra. Our approach uses an integration by parts argument under very mild conditions on the polynomial corresponding to the PDE of high order.

A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is presented. As usually, this a simple consequence of a similar theory for analogous families of basic objects, to which the main portion of this article is devoted.

In this paper, we prove some uniform estimates between Lebesgue and Hardy spaces for operators closely related to the multilinear paraproducts on $\mathbb{R}^d$. We are looking for uniformity with respect to parameters, which allows us to disturb the geometry and the metric on $\mathbb{R}^d$.

We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process.

For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical \textit{linear or bilinear adjoint restriction conjectures} for such functions and verify the \textit{linear adjoint restriction conjecture} for the paraboloid. We also interpret the restriction estimates in terms of solutions to the Schrödinger equation and establish the analogous results when the paraboloid is replaced by the lower third of the sphere.