Jean Bourgain’s research while affiliated with Institute for Advanced Study and other places

We prove convergence in norm and pointwise almost everywhere on $L^p$ , $p\in (1,\infty )$ , for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow-Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

70th birthday conference for Haim Brezis, Paris 2014

We show that the discrete Hardy–Littlewood maximal functions associated with the Euclidean balls in $\mathbb Z^d$ with dyadic radii have bounds independent of the dimension on $\ell ^p(\mathbb Z^d)$ for p ∈ [2, ∞].

We extend the decoupling results of the first two authors to the case of real analytic surfaces of revolution in ${\mathbb R}^3$. New examples of interest include the torus and the perturbed cone.

We present three applications of the Siegel mass formula, using the explicit upper bounds for densities derived in Bourgain and Demeter (Int Math Res Not 2015(11):3150–3184, 2014).

By using an idea of localized Galilean boost, we show that the data-to-solution map for incompressible Euler equations is not uniformly continuous in Hs(Rd)${H^s({\mathbb{R}}^d)}$, s≥0${s \ge 0}$. This settles the end-point case (s = 0) left open in Himonas–Misiołek (Commun Math Phys 296(1):285–301, 2010) and gives a unified treatment for all Hs. We also show the solution map is nowhere uniformly continuous.

We extend the decoupling results of the first two authors to the case of real analytic surfaces of revolution in $\mathbb{R}^3$. New examples of interest include the torus and the perturbed cone.

For the d-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space $H^s(\mathbb{R}^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore conjecture. In the previous paper [2], we introduced a new strategy (large lagrangian deformation and high frequency perturbation) and proved strong ill-posedness in the critical space $H^1(\mathbb{R}^2)$. The main issues in 3D are vorticity stretching, lack of $L^p$ conservation, and control of lifespan. Nevertheless in this work we overcome these difficulties and show strong ill-posedness in 3D. Our results include general borderline Sobolev and Besov spaces.