Related papers: Singular Integrals Along N Directions in R^2

We prove that the maximal operator obtained by taking averages at scale 1 along $N$ arbitrary directions on the sphere, is bounded in $L^2(\R^3)$ by $N^{1/4}{\log N}$. When the directions are $N^{-1/2}$ separated, we improve the bound to…

We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an…

We develop almost-orthogonality principles for maximal functions associated with averages over line segments and directional singular integrals. Using them, we obtain sharp $L^2$-bounds for these maximal functions when the underlying…

In a prior work [Hilbert transform along smooth families of lines math.CA/0310345] the authors introduced a variant of the Kakeya maximal function associated with Lipschitz maps from the plane into the unit circle. In this paper, we improve…

We construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N but the union of the doubles has area log log N/ log N. In…

We reprove Wolff's $L^{\frac{5}2}-$ bound for the $\R^3-$Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on…

We shall verify the Kakeya (Nikodym) maximal operator $K_{N}$, $N\gg 1$, is bounded on the variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^2)$ when the exponent function $p(\cdot)$ is $N$-modified locally log-H\"{o}lder continuous and…

Lebesgue space bounds $L^{p_1}({\mathbb R}^1) \times L^{p_2}(^1) \to L^q({\mathbb R}^1)$ are established for certain maximal bilinear operators. The proof combines a trilinear smoothing inequality with Calder\'on-Zygmund theory. A reference…

Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this direction set. Our main result provides an essentially sharp uniform bound,…

We show that for any dimension $d\ge3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in $\mathbb R^d$ for $d\ge3$ without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional…

We prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lacunary type thus extending to this setting, the works of Cordoba and of Nagel, Stein and Wainger.

In this article we obtain $L^2$ bounds for strongly singular integrals along curves in $\R^d;$ our results both generalise and extend to higher dimensions those obtained by Chandarana in the plane. Moreover, we show that the operators in…

Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.

This is an expository paper on the characterization of the even (or odd) smooth homogeneous convolution Calder\'on-Zygmund operators in R^n such that the maximal singular integral can be controlled in the L^2 norm by the singular integral.…

We discuss the L^p-boundedness of maximal singular integrals in the plane over a finite set V of N directions. Logarithmic bounds are established for a set V of arbitrary structure in the 2<=p<infinity range. Sharp bounds are proved for…

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the…

Let $\Omega $ be any set of directions (unit vectors) on the plane. In this paper we study maximal operator of the one dimensional maximal function computed in the directions of $\Omega$ We are interested in extensions of lacunary sets of…

We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds showing sharpness are proved using Kakeya set constructions. As a special case of…

We provide a condition on a set of directions $\Omega \subset \mathbb{S}^1$ ensuring that the associated directional maximal operator $M_\Omega$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. The techniques of proof…

We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this…