Related papers: $L^2$-Maximal functions on graded Lie groups

On $\mathbb{R}^n,$ a classical result due to Bourgain establishes the restricted weak $(\frac{n}{n-1},1)$ inequality for the full maximal function $M_F^{d\sigma}$ associated to the spherical averages. In this work we present an extension to…

The aim of my PhD work is to study the $L^p$-boundedness of operators on two classes of two-step nilpotent Lie groups, using Plancherel formulas and spherical functions as tools. The first class of groups consists of the groups of…

Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C. Calder\'on, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*}…

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the L^q-norm of the restriction of the Fourier transform of a function f in L^p (say, on Euclidean space) to a…

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct…

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into…

This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction…

In this paper, we introduce a criterion for maximal operators associated with Fourier multipliers to be bounded on $L^p(\mathbb{R}^d)$. Noteworthy examples satisfying the criterion are multipliers of the Mikhlin type or limited decay which…

Bourgain proved that the maximal operator associated to an analytic vector field is bounded on $L^2$. In the present paper, we give a geometric proof of Bourgain's result by using the tools developed by Lacey and Li.

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain's level set result on Strichartz estimates associated with Schr\"odinger equations on torus. Some sharp estimates on…

A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for k \geq k_{cr}(d), where k_{cr}(d) solves 2^{k_{cr}-1}+k_{cr} = d, every (d,k) set has positive Lebesgue measure. We give a short…

We investigate the $L^p$ mapping properties of maximal functions associated with analytic hypersurfaces in $\mathbb R^d$, with a particular emphasis on the role of transversality. Around points that are not transversal, we show that the…

We provide quantitative weighted estimates for the $L^p(w)$ norm of a maximal operator associated to cube skeletons in $\mathbb{R}^n$. The method of proof differs from the usual in the area of weighted inequalities since there are no…

Let $(X,d,\mu)$ be a metric space with doubling measure and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. We assume that there exists an $L$-harmonic function $h$ such that the…

We consider the $L^p$ mapping properties of maximal averages associated to families of curves, and thickened curves, in the plane. These include the (planar) Kakeya maximal function, the circular maximal functions of Wolff and Bourgain, and…

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse…

The purpose of this paper is to investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels. The principal results bring to light the intimate connection between…

In this paper, we investigate $L^p-$boundedness of the bilinear spherical maximal function associated with a general set $E\subset\R_+$. We quantify the range of $L^p-$boundedness in terms of a dilation-invariant notion of upper Minkowski…

In this paper we deal with lacunary and full versions of the spherical maximal function on the Heisenberg group $\mathbb{H}^n$, for $n\ge 2$. By suitable adaptation of an approach developed by M. Lacey in the Euclidean case, we obtain…