相关论文: A recursive bound for a Kakeya-type maximal operat…

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and…

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…

If the non-commutative L p space of SLn(Z) has the completely bounded approximation property for some non-trivial value of p, then some form of the Kakeya conjecture holds in dimension d, for all d $\le$ n+1 2 . The proof relies on a…

Let $K$ be a standard H\"older continuous Calder\'on--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed…

We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First, we obtain an $H^1$ to $L^{1,\infty}$ bound for lacunary maximal operators under a dimensional assumption on the…

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…

Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $\lambda$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($\lambda$ [G]) p MG p p where MG is the Hardy-Littlewood…

In this paper, we study the spherical maximal operator $ M_E $ over $ E\subset [1,2]$, restricted to radial functions. In higher dimensions $ d\geq 3$, we establish a complete range of $ L^p-$improving estimates for $ M_E $. In two…

We introduce a generalized inverse Gaussian setting and consider the maximal operator associated with the natural analogue of a nonsymmetric Ornstein--Uhlenbeck semigroup. We prove that it is bounded on $L^{p}$ when $p\in (1,\infty]$ and…

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing…

We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal multiplier operator and $\Psi$ is the sum of…

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…

In this paper, we investigate $L^p$ bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees $L^p$ bounds…

We investigate the existence of a maximiser among open, bounded, convex sets in $\R^d,\,d\ge 3$ for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if $d=2$), with constraints involving Lebesgue measure or…

Let $P(D)$ be the Laplacian $\Delta,$ or the wave operator $\square$. The following type of Carleman estimate is known to be true on a certain range of $p,q$: \[ \|e^{v\cdot x}u\|_{L^q(\mathbb{R}^d)} \le C\|e^{v\cdot…

A Kakeya set in $\mathbb{R}^n$ is a compact set that contains a unit line segment $I_e$ in each direction $e \in S^{n-1}$. The Kakeya conjecture states that any Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension $n$. We consider a…

We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, $k$-skeletons in $\mathbb{R}^n$. Although these operators are known not to be bounded on any $L^p$, we obtain…

Let n(2,k) denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset 2A contains {0,1,2,...,n-1}. A classical problem in additive number theory is to find an upper bound for n(2,k). In this…

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.

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical…