Related papers: Bourgain's condition, sticky Kakeya, and new examp…

In this paper, we first generalize the work of Bourgain and state a curvature condition for H\"ormander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for…

We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some…

We show that for each odd integer $n\ge 3$, there is an open dense subset of H\"ormander phase functions in $\mathbb{R}^n$ for which the associated curved Kakeya sets have Hausdorff dimension at least $\frac{n+1}{2} + d_n$ for some positive…

In this paper, we study curved Kakeya sets associated with phase functions satisfying Bourgain's condition. In particular, we show that the analysis of curved Kakeya sets arising from translation-invariant phase functions under Bourgain's…

Bourgain's discretization theorem asserts that there exists a universal constant $C\in (0,\infty)$ with the following property. Let $X,Y$ be Banach spaces with $\dim X=n$. Fix $D\in (1,\infty)$ and set $\delta= e^{-n^{Cn}}$. Assume that…

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made…

We consider Carleson-Sj\"{o}lin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of H\"{o}rmander-type oscillatory integral operators. We obtain improved…

This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin neighborhoods of line segments that point in different…

We consider a class of H\"ormander-type oscillatory integral operators in $\mathbb{R}^n$ for $n \geq 3$ odd with real analytic phase. We derive weak conditions on the phase which ensure $L^p$ bounds beyond the universal $p \geq 2 \cdot…

It is shown that $SL_2$ Besicovitch sets of measure zero exist in $\mathbb{R}^3$. The proof is constructive and uses point-line duality analogously to Kahane's construction of measure zero Besicovitch sets in the plane. A corollary is that…

Kakeya sets are compact subsets of $\mathbb{R}^n$ that contain a unit line segment pointing in every direction. The Kakeya conjecture states that such sets must have Hausdorff dimension $n$. The property of stickiness was first discovered…

Oscillatory integral operators with $1$-homogeneous phase functions satisfying a convexity condition are considered. For these we show the $L^p - L^p$-estimates for the Fourier extension operator of the cone due to Ou--Wang via polynomial…

We give a detailed outline of the proof that the Kakeya conjecture follows from the sticky case. This proof is due to Wang and Zahl and appears in a recent paper. The sticky case was proven in earlier work of Wang-Zahl, building on an…

We provide the final step in the resolution of Bourgain's slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq…

We apply the Bennett-Carbery-Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L^p estimates in the Stein restriction problem for dimension at least…

In the early 1990s, J.Bourgain proved a general result $K$-closedness result in the context of classical harmonic analysis. In this paper, we extend Bourgain's method to the semicommutative setting, making use of the recent semicommutative…

We establish new linear and trilinear bounds for collections of tubes in $\mathbb{R}^4$ that satisfy the polynomial Wolff axioms. In brief, a collection of $\delta$-tubes satisfies the Wolff axioms if not too many tubes can be contained in…

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild…

We investigate some bounded linear operators T on a Hilbert space which satisfy the condition |T | less or equal to |ReT |. We describe the maximum invariant subspace for a contraction T on which T is a partial isometry to obtain that, in…

The Rayleigh Conjecture for the bilaplacian consists in showing that the clamped plate with least principal eigenvalue is the ball. The conjecture has been shown to hold in 1995 by Nadirashvili in dimension $2$ and by Ashbaugh and Benguria…