Related papers: Lacunary maximal functions on homogeneous groups

We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the H\"older relation $1/p+1/q=1/r$. The boundedness region that we get contains at least the interior of…

We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in $L\log\log\log L(\log\log\log\log…

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this…

We prove maximal ergodic theorems for spherical averages on the Heisenberg groups acting on $L_p$ spaces over measure spaces not necessarily commutative, that is, on noncommutative $L_p$ spaces. The scale of $p$ is optimal in the reduced…

Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $\mu_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*\mu_t|$. We prove…

Bourgain in his seminal paper [2] about the analysis of maximal functions associated to convex bodies, has estimated in a sharp way the $L^2$-operator norm of the maximal function associated to a kernel $K\in L^1,$ with differentiable…

In this article, we study the fractional spherical maximal function and its lacunary counterpart. We study the necessary and sufficient conditions for $L^p-L^q$ boundedness of both maximal functions. In particular, we prove the restricted…

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp $L^p$-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full…

A few years ago, Bourgain proved that the centered Hardy-Littlewood maximal function for the cube has dimension free $L^p$-bounds for $p>1$. We extend his result to products of Euclidean balls of different dimensions. In addition, we…

We establish $L^{p_1}\times\cdots\times L^{p_k}\to L^r$ and $\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ type bounds for multilinear maximal operators associated to averages over isometric copies of a given non-degenerate $k$-simplex…

Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak g=\mathfrak g_1\oplus\mathfrak g_2$, where $[\mathfrak g,\mathfrak g]\subset \mathfrak g_2$. We consider maximal functions…

We establish the $L^p(\mathbb{R}^3)$ boundedness of the helical maximal function for the sharp range $p>3$. Our results improve the previous known bounds for $p>4$. The key ingredient is a new microlocal smoothing estimate for averages…

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…

Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_\Lambda f(n) := \sup_{\lambda \in \Lambda} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2\pi i \lambda p}}{p} |,\] where…

The optimal $L^p \to L^q$ mapping properties for the (local) helical maximal function are obtained, except for endpoints. The proof relies on tools from multilinear harmonic analysis and, in particular, a localised version of the…

We initiate the study of the $\ell^p(\mathbb{Z}^d)$-boundedness of the arithmetic spherical maximal function over sparse sequences. We state a folklore conjecture for lacunary sequences, a key example of Zienkiewicz and prove new bounds for…

We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in $L^{p}(\mathbb{R}^{n})$, $n\geq2$, where the radii of the spheres are restricted to a compact interval in $(0,\infty)$. These bounds…

We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously…

We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines…

We consider the $L^p \rightarrow L^p$ boundedness of a Nikodym maximal function associated to a one-parameter family of tubes in $\mathbb{R}^{d+1}$ whose directions are determined by a non-degenerate curve $\gamma$ in $\mathbb{R}^d$. These…