An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equations (NLS) on ${\mathbb{R}}^d$ with random and rough initial data. By working in the framework of $L^p({\mathbb{R}}^d)$ spaces, $p>2$, we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.     

Cauchy transform and Poisson’s equation

Let $u\in W^{1,p}\cap W_0^{1,p}$, $1?p?\infty$ be a solution of the Poisson equation $\Delta u=h$, $h\in L^p$, in the unit disk. We prove ${6?u6}_{L^p}?a_p{6h6}_{L^p}$ and ${6?u6}_{L^p}?b_p{6h6}_{L^p}$ with sharp constants $a_p$ and $b_p$, for $p=1$, $p=2$, and $p=\infty$. In addition, for $p>2$, with sharp constants $c_p$ and $C_p$, we show ${6?u6}_{L^{\infty }}?c_p{6h6}_{L^p}$ and ${6?u6}_{L^{\infty }}?C_p{6h6}_{L^p}$. We also give an extension to smooth Jordan domains.These problems are equivalent to determining a precise value of the $L^p$ norm of the Cauchy transform of Dirichlet’s problem.     

New inversion formulas for Radon transforms on affine Grassmannians

We obtain new inversion formulas for the Radon transform and the corresponding dual transform acting on affine Grassmann manifolds of planes in ${\mathbb{R}}^n$. The consideration is performed in full generality on continuous functions and functions belonging to $L^p$ spaces.     

On the Komlós,Major and Tusnády strong approximation for some classes of random iterates

The famous results of Komlós, Major and Tusnády (see Komlós et al., 1976 [15] and Major, 1976 [17]) state that it is possible to approximate almost surely the partial sums of size $n$ of i.i.d. centered random variables in ${\mathbb{L}}^p$ ($p>2$) by a Wiener process with an error term of order $o\left(n^{1∕p}\right)$. Very recently, Berkes et al. (2014) extended this famous result to partial sums associated with functions of an i.i.d. sequence, provided a condition on a functional dependence measure in ${\mathbb{L}}_p$ is satisfied. In this paper, we adapt the method of Berkes, Liu and Wu to partial sums of functions of random iterates. Taking advantage of the Markovian setting, we shall give new dependent conditions, expressed in terms of a natural coupling (in ${\mathbb{L}}^{\infty }$ or in ${\mathbb{L}}^1$), under which the strong approximation result holds with rate $o\left(n^{1∕p}\right)$. As we shall see our conditions are well adapted to a large variety of models, including left random walks on $G{L}_d\left(\mathbb{R}\right)$, contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We also provide some examples showing that our ${\mathbb{L}}^1$-coupling condition is in some sense optimal.     

Global weak solutions to the Vlasov–Poisson–BGK system for initial data in Lp(R3×R3)

We prove the existence of a global nonnegative weak solution to the Cauchy problem of the Vlasov–Poisson–BGK system for initial datum having finite mass and energy and belonging to $L^p({\mathbb{R}}^3\times {\mathbb{R}}^3)$ with $p>3$.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

Lp solutions of backward stochastic differential equations with jumps

Given $p\in (1,2)$, we study ${\mathbb{L}}^p$ solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)$-variables. We show that such a BSDEJ with $p$-integrable terminal data admits a unique ${\mathbb{L}}^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.